THERMAL ANALYSIS OF PRESSURIZED
WATER REACTORS Third Edition
L. S. Tong
Tong & Associates, Inc.
Gaithersburg, Maryland
Joel Weisman Nuclear Engineering Program University of Cincinnati Cincinnati, Ohio
American Nuclear Society La Grange Park, Illinois USA
Library of Congress Cataloging-in-Publication Data
Tong, L. S. (Long-sun), 1 9 1 5Thermal analysis of pressurized water reactors / L.S. Tong, Joel Weisman. - 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-89448-038-3 1 . Pressurized water reactors-Thermodynamics. 2. HeatTransmission. I. Weisman, Joel, 1 928II. Title. TK9203.P7T6 1 996 95-45359 621 .48'34--dc20 CIP
ISBN: 0-89448-038-3 Library of Congress Catalog Card Number: 95-45359 ANS Order Number: 300028
Copyright © 1 996 American Nuclear Society 555 North Kensington Avenue La Grange Park, Illinois 60525 USA
All rights reserved. No part of this book may be reproduced in any form without the written permission of the publisher.
Production/ Design: Lorretta Palagi Composition: Parkwood Composition
Printed in the United States of America-First Printing
To the memory of Jay Weisman, a young man full of promise.
AMERICAN NUCLEAR SOCIETY
BOOK PUBLISHING COMMITTEE
Tunc Aldemir The Ohio State University
Roger W. Tilbrook, Chair Argonne National Laboratory
John A. Bernard, Jr. Massachusetts Institute of Technology
Nicholas Tsoulfanidis University of Missouri-Rolla
Winfred A. Bezella Argonne National Laboratory Robert A. Krakowski Los Alamos National Laboratory Robert L. Seale University of Arizona
Raymond Waldo Southern California Edison Company Lewis A. Walton B&W Nuclear Environmental Services
CONTENTS Preface to Third Edition, Nomenclature, Glossary,
Power Generation
Reactor Configurations,
111
1.2
xvi
xix
Chapter 1 1.1
Basic Concept (Conventional PWR),
1 .1.2
Other Reactor Concepts,
1 1.3
Advanced Reactors with Passive Safety Features,
1 1.4
Status of Various Concepts,
1.2.1
1.2.2 1.2.3
29
Power Distribution in Unperturbed Uniformly Loaded 30
Effect of Fuel Loading,
32
Effect of Control Rods, Water Slots, Voids, and Burnable Absorbers,
41
1.2.4
Nuclear Hot-Channel Factors,
1.2.5
Heat Generation Within Fuel Elements,
1.2.6
Distribution of Power Among Fuel, Moderator, and Structure,
1.5
24
30 Cores,
1.4
18
Steady-State Power Generation and Distribution in Conventional PWR Cores,
1.3
xv
48 52
55
Methods for Estimation of Steady-State Core Power in Conventional PWRs,
58
1.3.1
Pin-by-Pin Diffusion Computations,
1.3.2
Nodal Methods,
1.3.3
Effective Fast Group Methods,
58
60 66
Power Generation and Distribution in Pressure Tube-Type Cores, 1.4.1
Overall Power Distribution,
1.4.2
Power Generation Within Fuel Clusters,
68
1.4.3
Effect of Refueling Procedures,
70
72
Transient Power Generation and Distribution,
74
1.5.1
Power Distribution Following a Load Change,
1.5.2
Power Generation During Shutdown,
79
74
68
CONTE N TS
1.6
1.7
Heat Generation in Reactor Structure and Moderator, 1.6.1
Structure,
1 .6.2
Moderator,
Chapter 2 2.1
2.2
Fuel Elements
Fuel Element Characteristics,
2.5
2.1.1
Fuels and Their Thermal Properties,
Fuel Element Cladding and Assembly Designs,
Behavior of U02 Fuel Elements,
93 114
127
2.2.1
Mechanical Properties of U02,
2.2.2
Fuel Densification and Restructuring,
2.2.3
Fuel Swelling,
127 130
135
2.2.4
Fission Gas Release and Internal Pressure,
2.2.5
Thermal Resistance of Fuel and Fuel-Cladding Gap,
139
2.2.6
Mechanical Behavior of Cladding,
2.2.7
Accuracy of Fuel Element Property and Behavior
164
168
Steady-State Conduction Under Idealized Conditions, Heat Generation Within a Fuel Plate,
2.3.2
Heat Generation Within a Fuel Rod,
2.3.3
Annular Fuel Elements,
172
182 183
2.4.1
Rigid Pellet, No Fuel-Pellet Friction,
2.4.2
Rigid Pellet, Fuel-Pellet Friction Considered,
2.4.3
Behavior Considering Pellet Plasticity,
Fuel Performance,
169
169
Realistic Evaluation of Steady-State Temperatures, 184
187
190
196
2.5.1
Typical Reactor Operating Conditions,
2.5.2
Fuel Behavior,
196
197
2.5.3
Improved Fuel Rod Design,
2.5.4
Evaluation of Fuel Performance During Normal Operation,
2.6
93
2.1.2
2.3.1
2.4
87
88
Models, 2.3
87
Thermal Design Basis, References,
85
85
202
204
Transient Fuel Rod Behavior,
209
2.6.1
Fuel Rod Temperature Transients,
2.6.2
Cladding Behavior During Transients,
2.6.3
Fuel Melting,
References,
232
230
209 222
150
CON TE NTS
Chapter 3 3.1
Basic Flow Characteristics,
3.1.1
3.2
3.3
3.5
242
Transport Equations for Single-Phase Flow,
3.1.2
Single-Phase Flow Friction,
3.1.3
Simple Flow Transients,
3.1.4
Pump Behavior,
242
244
257
258
Reactor Component Hydrodynamics,
261
3.2.1
Velocity Profiles and Turbulence in Fuel Rod Bundles,
3.2.2
Flow in Plenums,
270
3.2.3
Flow-Induced Vibration,
3.2.4
Hydraulic Design of Control Rods,
Two-Phase Flow, 3.3.1
3.4
Hydrodynamics
274 282
283
Two-Phase Flow Structure (Flow Patterns),
3.3.2
Void Fraction,
3.3.3
Two-Phase Pressure Drop,
290 300
3.3.4
Basic Two-Phase Flow Modeling,
309
3.3.5
Sonic Velocity and Critical Flow,
319
3.3.6
Pump Behavior with Two-Phase Flow,
Flow Instability,
284
342
342
3.4.1
Nature of Various Flow Instabilities,
3.4.2
Effects of Various Parameters on Flow Instability,
3.4.3
Dynamic Instability Analysis,
3.4.4
Parallel Channel Instability,
350 356
3.4.5
Flow Pattern Instability,
3.4.6
PWR Instability Concerns,
3.4.7
Recommended Methods of Analysis,
Liquid-Vapor Separation,
343
359 359 361
362
3.5.1
Entrainment and Carry-Over,
3.5.2
De-Entrainment,
3.5.3
Carry-Under,
3.5.4
Flooding,
362
368
371
372
3.5.5
Primary Steam Separation,
3.5.6
Phase Separation in Branching Conduits,
References,
382
375 378
350
261
C O N T E N TS
Chapter 4 4.1
Heat Transfer and Transport
Heat Transport Assuming One-Dimensional Behavior,
4.1.1
Steady-State Enthalpy Rise Along a Closed Coolant Channel,
4.2
4.3
4.4
396
4.1.2
Transient Heat Transport,
4.1.3
Heat Transfer Via Natural Ccmvection,
Subchannel Analysis,
4.2.2
Porous Media Approach,
419
419 447
Forced Convection Heat Transfer in Single-Phase Flow,
456
Empirical Equations for Single-Phase Heat Transfer,
4.3.2
Surface Roughness Effects on Heat Transfer,
4.3.3
Turbulence Promoter and Grid Spacer Effects,
Boiling Heat Transfer, 4.4.1
Flow Boiling Heat Transfer,
The Boiling Crisis or Critical Heat Flux,
460 461
462 466
4.4.3
Effects of Crud Deposition,
4.4.4
Post-C HF Heat Transfer,
4.4.5
Recommended Approaches to Boiling Heat Transfer
Condensation,
497
499
512
514
Basic Models,
514
4.5.2
Reflux Condensatior:1,
4.5.3
Spray Condensation,
516 517
Direct Contact Condensation on Flowing Liquid Streams,
4.5.5
456
462
4.4.2
4.5.4
4.7
418
4.3.1
4.5.1
4.6
401
Heat Transport in the Presence of Interacting Channels, 4.2.1
Estimation,
4.5
395
518
Steam Condensation on Containment Surfaces,
Heat Transfer in Reactor System Components, 4.6.1
Heat Transfer in Steam Generators,
4.6.2
Pressurizer Heat Transfer,
519
519 519
528
Cooling of Structure and Solid Moderator,
530
4.7.1
Temperature Distributiom Wilthin Core Structure,
4.7 .2
Temperature Distributiom Wilthin a Solid Moderator,
References,
535
530 533
C O N T E NTS
Chapter 5 5.1
Reactor Thermal and Hydraulic Performance
Basic Thermal Design,
546
5.1.1
Thermal Design Limitations and Approaches,
5.1.2
Fuel Rod Design,
5.1.3
Preliminary Core Design,
5.1.4
Hot-Channel Factor Evaluation for Conservative Core
5.1.5
Effect of Anticipated Operating Occurrences (AOOs) and
Design,
552
557
Major Accidents on Core Design, 5.2
Analysis of Core Performance During Normal Operation,
563
Optimal System Performance,
5.2.2
Effect of Core Flow Arrangement on Core Performance,
5.2.3
Interaction of Thermal-Hydraulic Design with Other
563
569
Subchannel Analysis Methods for Core Thermal-Hydraulic Design,
572
5.3.1
Objectives of Subchannel Analysis Procedures,
5.3.2
Sequential or Chain Subchannel Procedures,
5.3.3
Single-Pass Computational Methods,
5.3.4
D N B Computations Using Subchannel Analysis Procedures,
5.4
567
5.2.1
Areas, 5.3
546
548
572 573
574
577
Stochastic Thermal Analysis Procedures,
582
5.4.1
Assessment of Correlation Accuracy,
5.4.2
Stochastic Design Methods Based on the Total D N B
5.4.3
Assessment of DNB Failure Probability at Operating
Ratio,
582
583
Conditions,
586
5.5
Computer-Based Analysis of Pressure Tube Reactor Cores,
5.6
Reactor Protection and Core Monitoring,
5.7
5.6.1
Safety Criteria,
5.6.2
Core Monitoring Instrumentation,
589 592
5.6.3
Core Protection,
5.6.4
Set-Point Analysis (Conventional PWRs),
5.6.5
Thermal Protection in Pressure Tube Reactors,
5.6.6
Alternative DNB Design Limit,
Advanced PWRs, 5.7.1
587
589
594 598
607
608
Design Objectives and Specific Goals,
608
606
568
CONT E N TS
5.8
5.7.2
Core Modifications in Advanced Design Reactors,
5.7.3
Primary System Component Modifications,
Future Design Directions, References,
Chapter 6 6.1
6.2
6.3
6.4
6.6
614
615
Safety Analysis
Safety Approaches,
618
6.1.1
Safety Philosophy,
6.1.2
Safety Criteria,
618
619
Accident Evaluation Approaches,
621
6.2.1
Thermal-Hydraulics Approaches,
6.2.2
Neutronic Modeling,
6.2.3
Computer Configurations,
621
623 627
Anticipated Operating Occurrences,
627
6.3.1
Loss-of-Flow Accident ( LO FA),
6.3.2
Undercooling Transients,
6.3.3
Reactivity and Power Distribution Anomalies,
6.3.4
Overcooling Transients,
6.3.5
628
Changes in Reactor Coolant Inventory,
631 633
634 634
Accidents Due to Secondary System Rupture (Steam-Line Breaks),
6.5
609
613
635
Large-Break Loss-of-Coolant Accident ( L BLOCA), 6 .5.1
System Behavior,
6.5.2
Regulatory Requirements,
6.5.3
Key LOCA Phenomena,
6.5.4
Fluid-Structure Interactions,
636
636 639 640 655
6.5.5
Evaluation Models for L BLOCA Analysis,
6.5.6
Best Estimate Computer Models for L BLOCA Analysis,
659
6.5.7
Initial Conditions for Conventional PWRs,
668
6.5.8
Long-Term Recovery Following LBLOCA,
668
6.5.9
L BLOCA in CA NDU Reactors,
670
Small-Break Loss-of-Coolant Accident (SBLOCA), 6 .6 .1
Breaks Actuating Accumulator Flow,
6.6.2
Medium-Size Small Breaks,
6 .6 .3
Very Small Size Breaks,
6.6 .4
Break Location Effects,
674 678
673
673
672
665
C O N T E NTS
6.7
6.8
6 .9
6.10
6.6.5
SBLOCA Analysis,
6.6.6
SBLOCA in CA NDU Reactors,
681
Anticipated Transients Without Scram,
681
6.7 .1
Regulatory Requirements,
6.7.2
Characteristics of ATWS Events,
6.7 .3
Loss-of-Feedwater ATWS,
6.7 .4
ATWS Analysis Procedures,
Severe Accidents,
681 682
682 684
685
6.8 .1
Overall Perspective,
6.8 .2
Severe Accident Causes,
6.8 .3
Severe Accident Sequences in Conventional PWRs,
6.8 .4
Severe Accident Sequences in CA N D U Reactors,
6.8 .5
Important Physical Phenomena in Severe Accidents,
6.8 .6
Computer Programs for Analysis of Severe Accidents,
6.8 .7
Severe Accident Management,
6.8 .8
Probabilistic Risk Assessment,
Vapor Container Analysis,
685 687 688 691 693
709 709
712
6.9 .1
Containment Systems,
6.9 .2
Licensing Analysis,
6.9 .3
712
Vapor Containment During Severe Accidents,
713
Safety Features of Advanced Designs,
720
722
6.10.1
Redesigned Conventional PWRs,
722
6.10.2
Hybrid Safety System Approach,
724
6.10.3
Approaches Placing Primary Rei iance on Passive Safety Systems,
6.11
679
725
Safety Analysis Techniques for Advanced Reactors, References,
Author Index Subject Index
735 740
728
727
707
PREFACE to the Third Edition
The basic objective of this book is to present the principles underlying the thermal and hydraulic design of pressurized water reactors. In addition, the empirical data, engineering properties, and computer techniques re quired for design, but not available in conventional handbooks, are pre sented or referenced. This book is intended to provide an overview for nuclear engineering graduate students and to serve as a reference for en gineers working in the nuclear power industry. This work is not intended to be a design manual. While the foregoing objectives are essentially the same as those moti vating preparation of the first (1 970) and second (1979) editions, it was believed that a new edition was needed because of the many advances and changes that have occurred since 1979. The extensive improvements that have occurred in both understanding the phenomena involved and in cal culational techniques are reflected in the substantial additions to the text. Since the previous editions of this monograph have apparently been found useful by a number of readers, the basic format of previous editions has been essentially retained. However, an additional chapter has been added to accommodate the many developments in the area of safety anal ysis. We hope the revisions have enhanced the usefulness of this monograph. August 1995
L. S. Tong Joel Weisman
NOMENCLATURE This book follows the procedure of defining pertinent symbols after each equation. However, there are a number of symbols so frequently used that they have not been defined at each location of use. It is these symbols that are defined in this list. a/, av,aTP =
=
A=
a1
=
o
0' = =
01 =
De = e=
E =
f= f= Ffi, FIi, F£I =
Fq, F
�H
FqE, FKH
=
=
gc =
g = G = h =
hg =
H =
Htg =
Hv =
liquid, vapor, and two-phase sonic velocities, respectively sonic velocity, length/ time flow area, length2 specific heats at constant pressure and volume, respectively, energy / mass deg diameter, length extrapolated diameter of reactor, length diffusion coefficient, length2 equivalent diameter = 4A/perimeter, length eddy diffusivity, areal time internal energy, energy / mass flux depression factor, dimensionless friction factor, dimensionless nuclear hot-channel factors for overall heat flux, radial heat flux variation, and axial variation, respectively overall hot-channel factors for heat flux and enthalpy rise, respectively engineering hot-channel factors for heat flux and enthalpy rise, respectively gravitational conversion factor, (mass/weight)(length/ time2) gravitational acceleration, length/time2 mass flux, mass/time area heat transfer coefficient, energy / time area deg gap conductance, energy/ time area deg enthalpy, energy/mass enthalpy change on evaporation, energy / mass enthalpy of saturated vapor, energy/mass
N O M E N C LATU R E
J = mechanical equivalent of heat, mechanical energy / thermal energy = thermal conductivity, energy/ time length deg = effective multiplication factor = = infinite lattice multiplication factor K = constant, form friction factor L = length L, L' = height and extrapolated height of reactor, respectively, length p = probability, dimensionless p = pitch, length p' = perimeter, length P = pressure, force / area P' = power, energy/time Pr = Prandtl number = cpj.1/k q = total heat production rate, energy/ time q ' = linear heat output, energy/ time length q" = surface heat flux, energy/ time length2 q �rit = critical heat flux, energy/ time length2 q'" = volumetric heat flux, energy/ time length3 Q = volumetric flow, volume/ time r,R = radius, length R = ideal gas constant Re = Reynolds number = DeC / j.1 5 = slip ratio, vapor velocity/ liquid velocity = time = standard normal variate, dimensionless T = temperature u,v = velocity components, length/ time ug/ = drift velocity, length /time v = specific volume, volume/mass V = velocity, length/ time w' = transverse-turbulent mixing flow, mass/ length time x,y,2 = distance in coordinate directions, variables X = quality = mass flow rate of vapor / total mass flow rate Z = axial distance, length
k kef kex keff -1 koo
t t1
N OM E N C LATUR E
aa' ab =
thermal expansion coefficients of cladding and fuel, length /length deg a = void fraction, porosity a = parameter in escape probability calculation "y = Poisson's ratio "y = c,)CV
KO =
j..L = j..L
p:: =
p =
(J
=
(J
(J
L =
l (La /D l ) l2 attenuation coefficient, length viscosity, mass/length time mean value density, mass/volume stress, force/ area standard deviation surface tension macroscopic cross section, length neutron flux, neutrons /area time
-
1
GLOSSARY OF ACRONYMS AO AOO ATWS BOL BWR CANDU CCFL CCI CHF CPR DNB DNBR ECCS EOL FCI HPSI IRWST LBLOCA LCD LCO LHGR LOCA LOFA LPSI LSS NDT NRC PCI PORV PRA PWR RCC RPS RSS SAFDL SBLOCA SDL VVER
axial offset anticipated operating occurrence anticipated transient without scram beginning of life boiling water reactor Canadian deuterium (reactor) countercurrent flooding limit core-concrete interaction critical heat flux critical power ratio departure from nucleate boiling DNB ratio emergency core cooling system end of life fuel-concrete interaction high-pressure safety injection in-containment refueling water storage tank large-break LOCA late core disassembly limiting condition of operation linear heat generation rate loss-of-coolant accident loss-of-flow accident low-pressure safety injection limiting safety setting nil-ductility temperature Nuclear Regulatory Commission pellet-cladding interaction power-operated relief valve probabilistic risk assessment pressurized water reactor rod cluster control reactor protection system root-sum-square safe acceptable design limit small-break LOCA statistical design limit PWR of Russian design
CHAPTER ONE
Power Generation
A pressurized water reactor (PWR) is a water-cooled nuclear reactor under sufficient pressure to limit steam generation at the core exit and where the large quantity of heat produced is transferred to a secondary system via a heat exchanger. In this chapter, we briefly delineate the major reactor sys tems falling under this definition and then describe, from a thermal ana lyst' S viewpoint, the basic power distributions found within these reactors. The purpose of this discussion is to indicate the nature of the phenomena involved and some of the approximations that can be made. 1 .1 1 .1 . 1 1 .1 . 1 .1
REACTOR CON F I G U RATI O N S Basic Concept (Conventional PWR) Overall System
The basic PWR concept refers to the original (conventional) design now used in the United States, Europe, and East Asia. Alternative concepts, which are less widely used, are described subsequently (Sec. In basic PWR design, coolant flowing through the core is cooled by heat exchange with a secondary fluid. The overpressure adequate to limit bulk boiling is maintained by an electrically heated pressurizer. In a typical re actor system (Fig. the primary coolant is circulated through the re actor core by one or more high-pressure pumps. It proceeds to the heat exchanger where steam is generated and is then pumped back to the reactor core inlet. In the secondary system, feedwater is evaporated in the heat exchanger. Saturated steam goes through the turbine where it gives up its
1.1.2).
1. 1 ),1,2
T H E RMAL A N A L YS I S OF PWRS
Power Leads
Control Rod Drives
Pressuri zer
�m Waste Di
Reactor Vessel
�osol\
L A�� �:� ���
I��:!��?�
�
Cooling Water
Reactor Coolant Pump
Fig. 1 . 1
Typical PWR system (from Ref. 1 ) .
energy. It is then condensed in the condenser and returned to the heat exchanger by boiler feed pumps. In commercial reactors built for central station power, the reactor core, contained in a large pressure vessel with a removable head, is moderated and cooled by light water. Shielding around the vessel maintains low neu tron flux levels at the primary loop components; the reactor vessel, primary loop components, and auxiliaries are located within a structure (Fig. that can contain the steam and fission products released in the highly un likely event of a large break in the primary piping. 3 The primary system coolant in a PWR is completely isolated from the turbine; this is referred to as an In the boiling water reactor (BWR), the steam pro duced in the core goes directly to the turbine; this is called a Thus, the indirect cycle has the advantage of providing a radioactively clean secondary system, but requires expensive heat exchangers. Such factors as lower contained volume and simpler control requirements of the PWR com pensate for this and make the system economically competitive. The earliest fully commercial Western PWRs produced only about MW of electrical power and the primary system operated at approximately
1.2)
indirect cycle.
direct cycle. 120
POW E R G E N ERAT I O N
Fig. 1 .2
System arrangement for PWR power plant [from Westingho use Engineer, 25, 5 , 145 (1965»).
2000 psi. By the 1970s, plant designs had evolved so that most conventional plants were designed to produce 1 000 to 1 300 MW of electrical power with the primary system operating at pressures of about 2250 psi.* Although several newer (1994) designs are still in the 1000-MW range, some advanced designs will only produce about 600 MW of electrical power. Although the major application for pressurized water reactors is the pro duction of central station power, they originated as a result of a desire to provide power for naval propulsion units. PWRs have been very successful in this role and continue to be extensively used by the military. 1 . 1 .1 .2
Core Configuration
To date, fuel elements for central station power plants have typically con sisted of partly enriched uranium dioxide encapsulated in metal tubes. The tubes are assembled into bundles and cooled by water flowing parallel to their axes. Figure 1 .3 shows a horizontal cross section of a typical reactor.4 The fuel assemblies are placed in a configuration approximating a right circular cylinder; the larger the reactor, the more closely a circular cross section is approached. The assemblies are retained in position by a core baffle attached to the core barrel, which is both the structural support for the baffle and the core support plate on which the fuel assemblies rest. The core barrel is surrounded by a thicker ring of metal designated as the ther mal shield. The shield and water gaps between the core and vessel ther malize and attenuate the fast neutron flux emanating from the core. In addition, the shield attenuates the gamma flux; this serves to keep thermal *Although the Russian VVER plants generating 1 000 MWe operate at 2275 psia, the earlier 440-MWe plants operate at only 1 800 psia.
T H E R MAL A N A L Y S I S OF PWRS
Core Ba ffle Therm al Shi eld
Part Length Co ntro I Rod s
112.50 o.d.---�o.tl ...... �-----1 1 4.50 i.d .--.---� 1 22.50 o.d.--------c ....----.. .....------ 1 3 2.00 i.d. -----.... P l a n Vi ew of Core 121 Fue I Asse mbl i e s _____
.
Fig. 1 .3
Cross section o f PWR reactor vessel [from Proc. Am. Power Conf., 30, 298 (1968)].
stresses, due to gamma and neutron heating, and total neutron exposure of the reactor vessel within acceptable limits. In American and West European designs, the fuel rods are assembled into large square arrays,* which are held in position by spring clips on egg crate grids spaced about ft apart along the length of the assembly (Fig. Fuel tubes are omitted in a number of 10cations and are replaced by
1.4).
2
*Note that the cores of the Russian VVER reactors utilize fuel assemblies in which the fuel rods are arranged in an equilateral triangular array. In addition, the VVER cores generally did not use rod cluster control bu t have utilized control elements designed as fuel assembly followers. When the control elements are fully inserted, a number of fuel positions are actually occupied by control assemblies.
POW E R G E N ERAT I O N
ROD CLUSTER CONTROL
FUEL ROD CONTROL ROD
_::::::=--_��
GRID
...... _ .. ........-
MI XING VANES
DIMPLE
BOTTOM NOZZLE
Fig. 1 .4
Fuel assembly structure (courtesy Westinghouse Electric Corp.).
6
T H E RMAL A N A L Y S I S OF PWRS
hollow guide tubes that provide the structural support needed to tie the assembly together. The top and bottom nozzles are attached to these guide tubes. The control rods are clusters of absorber rods that fit in to the hollow guide tube. Each control rod member of a cluster consists of a sealed stainless-steel tube filled with a neutron-absorbing material such as boron carbide or a silver-indium-cadmium alloy. All fuel assemblies are identical and, hence, each has hollow guide tubes, although only a fraction of the assemblies are under control rod positions. In some designs, four adjacent rods are omitted from the lattice and the resulting space filled by a single guide tube. This results in a smaller num ber of larger control rods. The guide tubes that do not contain control rods are most commonly filled with plugging clusters to prevent unnecessary coolant bypass through the tubes. In some cases, the guide tubes of a number of assemblies are filled with burnable poison rods. These rods can provide the additional reactivity hold-down needed at the beginning of the cycle when the fuel loading scheme used leads to high discharge burnups. In earlier designs, cruciform-shaped control rods, fitted into cutouts at the edges of the fuel assemblies, were used. To prevent large wa ter gaps when the control rods were withdrawn, the rods were fitted with nonab sorbing extensions. When the control rods were inserted, the extensions projected below the core, necessitating a longer reactor vessel. The individ ual water slots left by the newer cluster design are small and rod extensions are not needed . The vertical reactor vessel cross section (Fig. illustrates the structural arrangements and coolant flow path.s The core barrel is attached to the lower core support barrel, which is hung from a ledge near the top of the vessel. The upper core support plate and upper control rod shrouds are attached to the upper core support barrel, which fits closely within the lowest support barrel and is supported by it. Both upper and lower core support barrels are designed so they can be removed . Normally, the lower support barrel remains in place throughout plant life. The primary coolant enters through the inlet nozzle and flows down ward through the passages between the core barrel and reactor vessel. After turning in the plenum area at the bottom of the pressure vessel, it flows upward through the core and exits through outlet nozzles via close fitting adapters provided as part of the lower core support barrel. Core design is treated in detail in Chapter S and the thermal design parameters for a number of PWR reactors are tabulated in Table S.Y
1.5)
1 . 1 . 1 .3
Steam Generation
Heat transferred to the secondary fluid is used to generate steam. Conse quently, the heat exchanger where this occurs is called a steam generator.
POW E R G E N E RATI O N
POWER GENERATION
CONTROL DRIVE MECHANISMS-
CONTROL ASSEMBLY FULLY WITHDRAWN
L-4u.+-+--- CONTROL ASSEMBLY SHROUD
CALANDRIA
lJUl LET NOZZLE CONTROL E SHROUD
FUEL ASSEMBLY ALIGNMENT PLATE FUEL ASSEMBLY CORE SUPPORT BARREL CORE SHROUD CORE SUPPORT ASSEMBLY
FLOW SKIRT
Fig. 1 .5
Vertical cross section of large
PWR
vessel (courtesy Combustion Engineering).
8
T H E RMAL A N A L Y S I S OF PWRS
A number of steam generator configurations have been used and, in all, the primary coolant has been circulated through the tubes and the secondary fluid contained within the shell. The low corrosion rates required to limit primary loop contamination led to the use of high-nickel alloys for the tubes. Initially, 300-series stainless steel was used exclusively for this pur pose, but its sensitivity to stress corrosion led to the use of Inconel in many more recent units. Excessive corrosion rates have also been encountered with Inconel under some conditions. The difficulty appears to be overcome by replacing the phosphate water treatment, originally used for the sec ondary system, by all-volatile chemistry (ammonia for pH control, hydra zine for oxygen scavenging). However, in some units that operated with phosphate chemistry for an appreciable time, it was found that some cor rosion difficulties persisted even after changing to all-volatile chemistry. Fletcher and Malinowski6 and Styrikovich et review typical steam gen erator operating experience. Section of this text provides further dis cussion of the problems encountered in steam generator operation and the design changes devised to ameliorate these difficulties. Horizontal steam generator units are closest to non-nuclear boiler prac tice. Single-drum or reboiler units (Fig. where steam separation takes place in the same drum as the bundle, can be used. Units of this type are still used in the Russian VVER designs. However, capacity is limited by the maximum drum size that can be fabricated or shipped and by the achiev able circulation ratio. Higher capacity can be obtained by using multiple drum units such as those shown in Figs. 1 .7 and Here, steam is generated in the lower drum containing the tube bundle and a two-phase mixture flows upward through the risers to the upper drum where the
4.6.1 al? 1.6),
1.8.
Steam Separating Gear -
Secondar y-Steam Out lets
Primary-Water
In l et
t Primary-Water Outlet
Fig. 1.6
Norm al Water Level Seco ndary Feedwater
Horizontal U-tube steam generator (From Nucleonics, 19, 7, 71 (1961)].
POWE R G E N E RAT I O N
Separators
9
Secondary-Steam Ou tlets
Steam Drum
Prim ory Water Inlet
Fig. 1.7
Horizontal straight-tube multiple-drum steam generator [from Nucleonics, 19, 7 71 (1961)].
steam is separated. Primary separation takes place at the vortex created in the separators at the entrance to the drum; water is returned to the lower drum via the downcomers. Units of this type were used at Shippingport; the first PWR in commercial service. Although current (1995) U.s. and West European plants have abandoned horizontal steam generators, a Japanese design of a simplified PWR system has proposed returning to these units.8 Such units are stated to have the advantages of freedom from sludge buildup on tube plates and a better capability for withstanding seismic events.
t· Fig. 1 .8
Horizontal U-tube steam generator with contour shell and multiple steam drums [from Nllcleonics, 19, 7, 71 (1961»).
T H E RMAL A N A L Y S I S OF PW R S
10
Unitized vertical steam generators were originally developed to provide compact, high-efficiency units for marine propulsion systems. Nuclear steam generators of this type account for the majority of units installed in large central station plants through All these units have used a U tube bundle configuration. A cross-sectional view of a typical vertical natural recirculation unit is shown in Fig. The two-phase mixture produced in the tube bundle rises to the primary cyclone separators. Each separator contains a helical pro peller, which establishes a centrifugal field that produces a free vortex. Wet steam exits through the central port and enters the secondary steam sepa rator in the steam drum, while the separated water flows down around the cyclone separators. Feedwater is added through a ring just above the tube bundle; the subcooled water flows downward in the space between the tube bundle and shell, completing the recirculation loop. Some of the later model steam generators are equipped with an integral preheater located on the cold leg of the tube bundle. Such a preheater in creases the secondary surface temperature in the boiling region and pro duces higher pressure steam. Feedwater is introduced into the preheater through a nozzle in the lower shell. The feed water flows through a specially baffled region before it joins the main recirculating flow.
1994.
1.9.
Seconda ry-Stea m Outl e t
Stea m Se pa rating Gea r
Ma nwa y �-1-1� 1I
Seconda r y Fe edwater
Tube ----tWJ../ Bund l e
Prima r y Water Ou t l et
Fig. 1.9
Prim a ry Water Inlet
Vertical U-tube single-drum steam generator [from Nucleonics, 19, 7, 71 (1961 )].
POW E R G E N E RATI O N
Steam generators for Canadian deuterium-uranium (CANDU) reactors are vertical U-tube units generally similar to the units used for vessel PWRs. In most cases, the steam separators are located in the upper section of the vertical unit. However, in the Bruce A generating stations, the outlets of two vertical U-tube steam generators are connected to a single horizontal separating drum. The desire to produce superheated steam has led to the development of the design. A sectional side view of such a unit9 is shown in Fig. To provide a comparison of relative size, the outline of a vertical natural-circulation boiler designed for the same conditions is superimposed on it. The reactor coolant enters the once-through generator at the upper plenum, flows through Inconel tubes, and exits from the lower plenum. The entering feedwater is sprayed from an annular ring and mixed with steam bled from the upper end of the boiler section. The mixture flows down the annular down comer and into the tube bank. No distinct water level is present since the mixture quality gradually increases to and superheating starts. The superheater region is baffled to obtain the cross flow and high velocities needed for efficient heat transfer to steam. The superheated steam exits through outlet nozzles near the upper end of the shell.
once-throu 1.10. gh
100%
Primary I n let
Steam Outlet Feedwater I n let
Downcomer An n u lus
Superheater Baffles Steam Ports
133-in. d i ame ter
62 ft 8 in. In conel Tubes
Primary Outlet
Fig. 1.10
Once-through steam generator; recirculating unit shown in outline [from Ref. 9].
T H E RMAL A N A L Y S I S OF PWRS
1 . 1 . 1 .4
Other Primary System Components
In addition to the reactor vessel and steam generators, the primary system contains a pressurizer, large reactor-coolant circulating pumps, and con necting piping. In the most common arrangement, each coolant circulating loop contains a steam generator and pump. In modern designs based on the use of a vertical U-tube steam generator, a medium size plant (e.g., 600 MWe) would contain two loops whereas a large plant (e.g., 1000 MWe) would contain four such loops. The pumps are located on the cold legs (primary fluid outlet lines from the steam generators). In plants using once through steam generators, the outlet plenum of each steam generator is connected to two circulating pumps. A large PWR of this type would have four circulating pumps, but only two steam generators. The early Russian reactor designs utilized relatively small horizontal steam generators. These plants, with a maxinlum capacity of 440 MWe, re quired six loops. Later designs have used larger horizontal steam genera tors and the 1 000-MWe VVER requires only four loops. Primary Coolan t Pumps. In a reactor system for submarine propulsion, leakage from the primary system cannot be tolerated. Because all mechan ical pump seals allow at least some small leakage, canned motor pumps, which have no shaft seals, are used. In such a pump, the entire pump and driving motor are completely enclosed within a pressurized casing. To keep the motor windings dry, the stator and rotor are canned within thin sheets of nonmagnetic material (i.e., 300-series stainless steel). An auxiliary im peller circulates a small amount of water in the narrow space between the stator and rotor liners so as to (1) provide lubrication for the journal and thrust bearing and carry away motor heat. This coolant exits through a tube that is wrapped around the motor casing. External cooling water is circulated over this coil in the space between the coil and an outer jacket. The rotor is directly connected to the pump impeller. A labyrinth type seal limits the interchange between the fluid being pumped and the fluid in the space between the stator and rotor. Since the first central station PWRs were derived from naval reactor de signs, the early PWRs used canned motor pumps. However, leakage re quirements in a central station plant are much less stringent than for a naval reactor and mechanical seals with low leakage rates are acceptable. Since large pumps with mechanical seals are both less expensive and more effi cient than canned motor pumps, all second and subsequent generation plants in the United States and Western Europe used mechanical seals.*
(2)
"Both first- and second-generation VVER plants used canned rotor pumps. However, the third generation VVER 1 000 plants employ shaft-sealed pumps. These shaft-sealed pumps are at tached to a flywheel to provide extended flow coastdown in case of a power loss.
POWER G E N E RATI O N
13
In the usual main circulating pump design (Fig. 1 .1 1 ), high-pressure in jection water is used to keep the major leakage through the seal inward rather than outward. A typical 90,000-gpm pump requires of the order of 8 gpm of injection water. About 5 gpm flows downward along the shaft removing heat conducted by the shaft and parts of the thermal barrier. The remaining injection flow passes upward through and around the water lubricated radial bearing of the seal assembly. A typical shaft seal consists of three face seals operating in a series ar rangement. Although the leakage rate for the first (primary) seal may be as high as 3 gpm, the leakage rate from the third (outermost) seal is designed to be of the order of 1 00 cm3/ hr of injection water at operating conditions. Each of the four pumps for a 900-MWe plant would have a capacity of about 90,000 gpm and be operated by a 9000-hp induction motor. The pumps for a 1 300-MWe unit would each have a capacity of about 1 1 0,000 gpm and be operated by a 12,500-hp motor. The absence of high-pressure injection water leads to pump seal failure. Such a failure produces a small loss-of-coolant accident (LOCA). In view of this possibility, some advanced PWR designs (1 994) call for the use of canned motor pumps. Pressurizer. The pressurizer is placed on a pipeline connected to the hot leg of one of the primary circulation loops. A cutaway drawing of a typical unit is shown in Fig. 1 . 12. During steady-state operations, approximately 55 to 60% of the pressurizer volume is occupied by water and 40 to 45% by stearn. Electric immersion heaters, located in the lower section of the vessel, keep the water at saturation temperature and maintain a constant system operating pressure. When the volume of the water in the system increases, the entry of the primary coolant into the pressurizer raises the water level and compresses the stearn. The attendant increase in pressure actuates the valves in the spray line. Cold-leg reactor coolant from the outlet side of the pumps then sprays into the stearn space and condenses a portion of the stearn. In the face of a coolant volume contraction, water from the pressurizer flows outward, thus reducing the pressurizer level and pressure. At the same time, some of the water in the pressurizer flashes to stearn. In addi tion, the immersion heaters corne on and further limit the pressure reduction. A power-operated relief valve is connected to a nozzle at the top of the pressurizer to protect against pressure transients that are beyond the ca pacity of the pressurizer. In such a case, the relief valve discharges stearn into a pressurizer relief tank, which is partly filled with water and is where the stearn condenses. If the system pressure were to continue to rise, a spring-loaded safety valve would open and thus prevent system failure.
T H E RMAL A N A L Y S I S O F PWR S
14
FLYWHEEL
UPPER RADIAL BEARING THRUST BEARING
MOTOR SHAFT
THRUST BEARING OIL LIFT PUMP
MOTOR STATOR
+ MOTOR
MAIN LEAD
MOTOR UNIT
CONDUIT BOX
ASSEMBLY
LOWER RADIAL BEARING SEAL HOUSING
NO.
3 SEAL
NO. I SEAL LEAK OFF
LEAK OFF
MAIN FLANGE NO.
2 SEAL
LEAK OFF
COOLING WATER OUTLET
PUMP SHAFT COOLING WATER INLET
RAD I AL BEARI NG
ASSEMBLY
THERMAL BARRIER AND DISCHARGE NOZZLE
SUCTION
HEAT EXCHANGER
CASING
IMPELLER
HOZZLE
Fig. 1.11
Cutaway of typical reactor coolant pump (courtesy Westinghouse Electric Corp.).
15
POWE R G E N E RATI O N
RELIEF NOZZLE SPRAY NOZZLE
MANWAY UPPER HEAD
� INSTRUMENTATION NOZZLE
LIFTING TRUNNION (LOAN BASIS)
HEATER SUPPORT PLATE
LOWER HEAD INSTRUMENTATION NOZZLE ELECTRICAL HEATER
SUPPORT SKIRT SURGE NOZZLE Fig. 1.12
Cutaway of typical pressurizer (courtesy Westinghouse Electric Corp.).
T H E RMAL A N A L Y S I S OF PWRS
16
A typical pressurizer for a 1 300-MWe plant would be about 41 ft long, 10 ft in diameter, and have a 5- to 6-in.-thick shell. The electric heater ca pacity would be about 1 .75 MW. 1.1 . 1 . 5
A uxiliary Systems
In addition to the primary loop, several auxiliary systems are necessary for the proper functioning of the plant during normal operation and to provide adequate core cooling during shutdown and accident situations. The major systems and their functions are as follows:
2.
Chemical and volume control system: Residual heat removal system:
3.
Safety injection system:
1.
4. 5.
Fills and pressurizes system, main tains water level, reduces concentration of contaminants, adjusts con centration of chemical poisons added to primary coolant. Transfers thermal energy from the reactor coolant during shutdown and refueling operations. System used in conjunction with safety systems during LOCA to remove heat from water being recirculated by injection system. Rapidly injects water from gas pressurized ac cumulators during early phase of large LOCA, adds borated water from high head pumps (needed during small LOCA where depres surization is slow), adds large volume of borated water from low head pumps during LOCA (needed during large LOCA with rapid depressuriza tion). Provides for fuel insertion and removal from the core, provides for fuel storage. Provides a vessel capable of containing the pres sures and temperatures generated by a large LOCA, provides a spray of cold water to limit the containment pressure and to remove iodine from the containment system, provides heat removal from the con tainment during normal and accident conditions.
Fuel handling system: Containment system:
1.1.1 .6
The Saturated Steam Turbine Cycle
Large, modern, fossil-fueled steam plants produce steam at pressures above 3200 psig and temperatures around 1 000 to 1 050°F. Steam conditions for PWR power plants are vastly different; the relatively low temperature of the primary system coolant, coupled with the necessity of providing an appreciable temperature difference across the heat exchangers, leads to steam pressures in the range of 400 to 950 psig (newer plants are at the upper end of the range). The majority of PWR plants produce dry and saturated steam. The main drawback of a saturated steam cycle is its excessive endpoint moisture (20 to 24%), which leads to blade erosion and blade efficiency losses that reduce thermal efficiency. Therefore, some method of moisture removal must be
POW E R G E N E RAT I O N
- ----
r-------r-
I I I I
--,
I -, I LI r --, I r--.lh....-Steam -.L_I I I
I I I I I
�
IL_,� I __ JI
I
L-r- ...J
I
I I I L... _______
I L......._--,I
Condenser
L
..., I I I
Pump
Drain Pump
Fig. 1.13
The
PWR
steam cycle.
used. Evaporation by constant temperature throttling reduces the steam pressure and is detrimental to cycle efficiency. However, the mechanical steam separator has proved to be efficient and desirable since such moisture separators both reduce endpoint moisture and improve turbine efficiency. Efficiency improvement is enhanced when moisture separation is com bined with reheating. Figure 1 . 1 3 shows the general arrangement · of the elements of a power generation cycle for a medium-size PWR using a com bination moisture separator/live-steam reheater, which is placed in the crossover piping between the high- and low-pressure turbines. By means of a finned tube heat exchanger, this device removes moisture and transfers heat from steam bypassed ahead of the throttle valve to the main stream. Since superheat increases the inlet temperature of a steam turbine and reduces endpoint moisture, it improves cycle efficiency considerably. In a closed cycle, superheating can be achieved by using either fossil fuel or the once-through steam generator previously described. The secondary steam from two early demonstration power reactors, the Carolina-Virginias Test Reactor (CVTR) and Indian Point I , was superheated by fossil fuel. How ever, the economics of fossil-fired superheat has not proved attractive and the more modern plants have used saturated steam or superheated steam from a nuclear steam generator. Since the primary heat source is at a rel atively low temperature, the amount of superheat that can be supplied by such a steam generator is low; i.e., on the order of 50°F. Thus, the steam
18
T H E RMAL A N A L Y S I S OF PWR S
cycle followed is quite similar to the dry and saturated cycle previously described. Moisture separation or moisture separation plus reheat is still required to prevent excessive endpoint moisture. The steam cycle for a large PWR is very similar to that shown in Fig. However, a double-flow intermediate pressure turbine would be lo cated before the two low-pressure units. An extensive discussion of thermodynamic cycles for nuclear power plants is provided by Kalafati, l O conventional equipment is described by Artusa, l l EI-Wakil,1 2 and Weisman and Eckart.1 3
1.11.
1 .1 .2 1.1.2.1
Other Reactor Concepts Compact, L ight-Water-Moderated Reactor Systems
A compact, low-weight system is of paramount importance in plants de signed for ship propulsion. One method 14 achieves this by enclosing the reactor core and steam generator within the same pressure vessel. Coolant flows up through the core and chimney and down through the heat ex changers; it exits through the outer annulus of the double pipes leading to the pumps. The pumps are closely coupled to the reactor, but outside the primary vessel for ease of maintenance. A pressurizer is eliminated by sim ply maintaining a steam space above the heat exchangers. Since this steam is in equilibrium with water at the average core exit temperature, the qual ity of the fluid at the hot channel exit is significant. A compact reactor design of this general type has operated successfully in the merchant ship Reactors of this type have also been pro posed for combined desalination and power production applications. In meeting the various military requirements, such as those for small plants in remote areas, weight and size can be of even greater importance. Here, highly enriched fuel can be used. Typical of this reactor group is the PM series of power plants that can be transported by air. The fuel elements are �O.5-in.-diameter hollow tubes made up of a highly enriched U02 stainless dispersion between stainless-steel claddings. The fuel is cooled by water flowing both inside and outside the fuel tubes. In PM-l (Ref. 1 5), the fuel tubes are assembled into six pie-shaped assemblies and one small cen tral assembly, which are held together by brazed ferrules and end plates. Because of high costs and safety considerations, there is essentially no current interest in compact plants for merchant ships or land-based military units. Of course, the military continues to use compact nuclear reactors for naval vessel power plants. In addition, the Russians operate icebreaker ves sels powered by such reactors. Proposals have also been made for com mercial submersible vessels powered by compact nuclear plants.
Otto Hahn.
1 .1 .2.2
Heavy- Water-Moderated Reactors
Although natural uranium alone cannot bring a light-water-moderated re actor to criticality, it is a satisfactory fuel for large reactors moderated and
POW E R G E N E RATI O N
19
cooled by heavy water. In areas such as Canada, where enrichment facilities are lacking but heavy water and natural uranium are available, attention has turned to this type of reactor. The simplest design has the moderator and coolant in the same circuit and is similar in many respects to a light water reactor (LWR). However, the greater distance (i.e., the greater slowing-down length) required to ther malize fission neutrons in heavy water requires a much greater moderator to-fuel ratio. One proposed method would achieve this by using fuel elements consisting of concentric rings with wide spacing between the rings and fuel elements. Alternatively, clusters of cylindrical fuel rods with wide spacing between clusters can be used. The large-size cores that result lead to reactor vessels far larger than those required for a light water plant of the same rating. The concept is most readily adapted to small- and medium sized reactors. The 65-MWth Agesta reactor16 in Sweden was an early ex ample of this concept. The Argentinian vessel type pressurized heavy water reactors (PHWRs) are more recent examples of this approach. In the vessel-type PHWR design}7 the large high-pressure reactor vessel contains a moderator tank, which is pierced by a large number of vertical fuel channels (a 320-MWe reactor would contain about 250 such channels, whereas a 750-MWe reactor would contain about 450 channels). A fuel as sembly running the full length of the core is inserted into each of the chan nels. A typical fuel assembly would contain 37 rods arranged on a triangular pitch. Because moderator and coolant are maintained at the same pressure, thin-walled zirconium alloy channels can be used. Holes at the upper end of the coolant channels interconnect the channels with an outlet plenum. However, the top of each channel is connected to a thick-walled pressure tube, which pierces the head of the vessel. This enables on-line refueling using an automated refueling machine. Such an arrangement is necessary because the natural uranium fuel allows average discharge burnups of only about 6000 MWd/tonne. By partially separating the moderating and coolant functions, the diffi culties imposed by the greater slowing-down length of heavy water can be surmounted. The fuel-containing channels are sufficiently separated to pro vide enough distance between them for neutron moderation. A further separation between moderating and coolant functions occurs in pressure-tube designs. In these designs, no large high-pressure reactor vessel is employed but the fuel clusters are contained within heavy-walled pressure tubes. These tubes must have high strength and good corrosion resistance. To maintain good neutron economy, the pressure-tube material must also have a low neutron absorption cross section. All reactors of this type have used zirconium alloy (usually zirconium with a small amount of niobium) pressure tubes to meet these requirements. Each pressure tube is insulated from the moderator by a narrow gas gap between the pressure tube and an outer tube of Zircaloy. This allows the moderator to be contained in a lar2:e low-pressure tank called a calandria.
T H E RMAL A N A L Y S I S OF PWR S
20
STEAM PI PES
PRESS U R I Z E R
EJ
o
� o
Fig. 1 . 1 4(a)
LI GHT WAT E R
STEAM
LIGHT WAT E R CONDENSATE
MODERATOR PUMP
HEAVY WATER COOLANT
HEAVY WATE R MODERATOR
CANDU reactor heat transport circuit [from P Gulshami, Trans. A m . Nucl. Soc.,
55, 459 (1987)].
In the early CVTR reactor,18 the calandria , c ontained vertical U-tubes so that all pressure tube connections were made at I t he top. In CANDU reactors, the pressure tubes are placed horizontally and are of the once-through type. 19 The CANDU reactors represent the only heavy-water-moderated pressure-tube design of major industrial importance. In the CANDU reactors, the fuel channels are generally arranged in one or two loops. Each lcmp, as shown in Fig. 1 .1 4(a), contains two pumps, two steam generators, two inlet headers, and two outlet headers. The fuel chan nels are connected in parallel to the headers so that they provide two passes through the core. The bidirectional flow that this provides enables the flow to be in opposite diI1ections in adjacent pressure tubes. This flow arrange-
POW E R G E N E RAT I O N
S h u t d o w n syste m 2
e •
V a l ve n o r m a l l y open V a l ve n o r m a l l y c l osed
Fig. 1 . 1 4(b)
CANDU control and shutdown arrangement [from Z. Domaratzki, Nllcl. Safety, 24, 452 (1983)].
R.
Atchinson, F. Bogdand,
ment, together with bidirectional refueling, avoids a reactivity tilt toward one end of the core. Control and shutdown of CANDU reactors are achieved by insertion of vertical control rods into the low-pressure moderator. A backup shutdown mechanism is provided by a boric acid injection system. The general ar rangement of these systems is shown schematically in Fig. 1 . 14. Note that only a small fraction of the total number of pressure tubes is shown in Fig. 1 .14(b). From a thermal-hydraulic point of view, some of the significant differ ences between the CANDU reactors and the basic PWR (conventional) de sign follow: 1 . Fuel elements placed in a hexagonal array instead of a square array 2. Coolant channels connected only at core exit and outlet instead of along en tire core length 3. Horizontal coolant flow instead of upward flow
T H E RMAL A N A L Y S I S OF PWRS
4. Primary system operation at pressures in the 1300- to 1400-psi range instead of the 2000- to 2250-psi range 5. Thermodynamic and transport properties of heavy water that differ slightly from those of light water 6. Core lengths of 19 to 20 ft in contrast to the 10 to 12 ft of conventional designs. The use of pellets of U02 produced from natural uranium as the fuel for all current (994) pressure-tube reactors leads to on-line refueling of CANDU reactors just as it did in the PHWR. Fresh fuel is added at one end of the channel and burned fuel is removed at the other end. Reactors fueled by thoria slightly enriched by 223U have been proposed. 20 If fabri cation and reprocessing losses can be held low, a self-sufficient cycle might be possible. That is, the bred 233U might be sufficient to replace the 233 U consumed in the operation. More recently,2 1 attention has shifted to the possibility of fueling CANDU cores with depleted (enrichment plant tails) U02 pellets containing small amounts of Pu02 (mixed oxide fuel). Another fuel cycle that has been examined uses U02 pellets obtained from the slightly enriched uranium chemically separated from discharged LWR fuel. 22 This reuse would improve overall utilization of uranium resources. Any of the advanced fuel cycles could also be used with a vessel-type PHWR. Use of oxide fuel with a fissile material content in excess of that of nat ural uranium would increase allowable burnup and thus decrease the number of fuel assemblies that must be fabricated and (2) waste disposal costs. However, the presence of plutonium would substantially increase the cost of fabricating an individual assembly. Similarly, the cost of fabrication of reprocessed light water fuel may be increased by the presence of trace quantities of fission products. The heavy-water-moderated reactors used for production of plutonium and tritium at the U.s. Savannah River Plant may also be considered as pressure-tube PWRs. Although they operate at low pressures and all the heat they produce is dissipated without any power production, these re actors do meet the definition of a PWR since no boiling occurs in the reactor. The reactor coolant is pressurized slightly and contained within vertical pressure tubes arranged in a low-pressure calandria. At present (992), only a single reactor is operated for tritium production. The pressure tubes of this reactor hold a series of concentric annular elements, which alternately contain fuel (U-AI clad in Al) and target material. In contrast to the upward flow design of conventional PWRs, the fuel elements here are cooled by a downward flow of heavy water.
0)
1 .1 .2 . 3
Graphite-Moderated Reactors
Graphite-moderated and light-water-cooled reactors have a long history in the production of plutonium. It is natural that this concept be adapted to
POWE R G E N ERATION
power generation by use of higher coolant temperatures. Reactors of this type that maintain an overpressure sufficient to prevent bulk boiling fit our definition of a PWR. The first atomic power station in the former Soviet Union was of this nature. The reactor is encased in a sealed cylindrical steel jacket filled with graphite bricks pierced by fuel channels; an atmosphere of helium or nitrogen is maintained to prevent graphite oxidation. The fuel element design23 was rather unusual: The active section of each fuel channel consisted of a graphite cylinder pierced by five tubes. The coolant flowed down the central tube, then upward through passages inside the annular fuel elements. The fuel elements consisted of slightly enriched hollow uranium-metal cylinders clad on both sides by thin stainless steel. The com plexity of the fuel element design makes this reactor ill suited for large central station application. A large, dual-purpose reactor (power and plutonium production) was erected at what was then the Hanford laboratory of the u.s. Atomic Energy Commission (AEC). The reactor, designated the NPR, consisted of a very large, 33- X 33- X 39-ft rectangular stack of graphite, penetrated by 1000 horizontal pressure tubes. 24 The 2.7-in.-Ld. Zircaloy pressure tubes con tained annular fuel elements of uranium metal clad in Zircaloy. If the re actor had been converted to power production alone, U02 could have replaced the uranium metal. Heat was removed from the NPR by six loops, each containing two steam generators. During dual-purpose operation, a portion of the low pressure steam proceeded to two turbines, while the remainder proceeded to 16 dump condensers. For power production only, the dump condensers could be eliminated and higher pressure steam could be generated. RMBK reactors, used in the former Soviet Union, have some points of similarity to the NPR design. The RMBK units are also graphite-moderated reactors in which the fuel elements are contained within pressure tubes and are cooled by light water. However, bulk boiling of the coolant occurs in the RMBK design and the reactor has substantially different operating char acteristics. Because the nuclear reactor that was the source of the disastrous accident at Chernobyl was of the RMBK design, concern was expressed about other water-cooled, graphite-moderated power reactors. Although analyses of the NPR system indicated that an accident like that at Chernobyl would not occur, the NPR has been shut down. �
1 . 1 . 2 .4
Supercritical Reactors
Reactors using supercritical water substance, or water above the critical pressure, as a reactor coolant, have been proposed for future applications. Such systems have the potential advantages of higher efficiency due to higher steam pressure and temperature, higher enthalpy rise through the core, avoidance of limitations imposed by boiling heat transfer, and lower contained energy in the system. Both direct and indirect cycle reactors were considered. Since no phase change can occur at supercritical pressures, both
24
T H E RMAL A N A LY S I S OF PWR S
cycles fall under our definition of a PWR. Supercritical reactors can be of the pressure vessel or pressure tube type. 2S Supercritical reactors require the successful development of a high temperature collapsed-cladding fuel element; cladding is so thin that it re lies on the fuel for its support. In addition, water chemistry conditions that will suppress radiolysis, control corrosion, and prevent deposition of solids on the fuel must be determined. There is no current (1 995) interest in such systems. 1 .1 .2.5
District Heating Plants
In addition to the use of nuclear power for the generation of electricity, appreciable consideration has been given to its use for district heating plants. Although the largest district heating demonstration unit has been a natural-circulation BWR, several PWR concepts have been proposed. The SLOWPOKE design26 uses a swimming-pool reactor with just an atmos pheric overpressure. However, since the reactor itself is under about 10 m of water, one might marginally consider it a PWR. SLOWPOKE is the only PWR concept for which a demonstration plant has been built. Other reactor concepts are more like conventional PWRs. The SECURE reactor is a loop-type PWR at 7 bar in a water pool. The GEYSER, HERE, and THERMOS designs27 are all of the so-called "integrated" type; that is, the steam generators and the reactor core are located in the same vessel. THERMOS uses forced circulation while the other integrated designs use natural circulation. Successful use of district heating reactors requires the reactors to be in proximity to a population center. Public acceptance of such a unit remains to be determined. 1 . 1 .3 1 . 1 .3.1
Advanced Reactors with Passive Safety Featu res Approaches Taken
As PWR technology has matured, the understanding of safety issues has advanced substantially. This has led to a gradual increase in the number of and complexity of the safety systems required to cope with all of the accident scenarios that can be hypothesized. Although the resulting level of public safety is believed to be very high, safety in the event of an accident does require the proper startup and operation of complex systems. Al though appropriate redundancy is used to minimize failure, this further complicates the reactor system. The desire to simplify the reactor system and to lessen dependence on active systems has led to the development of several new system designs. Tong28 has attempted to codify the require ments for such advanced designs and has developed a set of principles on which such designs should be based. It is hoped that such systems will lead
POW E R G E N ERAT I O N
to lower costs and decreased construction time as well as high safety levels with improved public acceptance. The advanced designs can be divided into two categories. The first of these has been termed Such reactors use the technology of current ( 1994) reactors but significant changes in plant design and layout are made. In the event of an accident, safety for most of these designs depends on passive systems and systems started by simple actions, such as the opening of a valve, but systems that are passive in operation (no moving equipment needed for continued safety system operation). A second category includes future reactors with radical changes in safety systems. Such reactors would have safety systems that require neither ac tive initiation nor active operation of equipment in order to function. Pro ponents of this approach have used the acronym PRIME29 (fassive safety, Besistant safety, Inherent safety, Malevolence resistance, gxtended time for external aid after an accident) to describe such units.
evolutionary technology reactors.29
1.1 .3.2
Evolutionary Technology Reactors
Reactor designs in this category generally have the following features30 : 1 . All water needed for heat removal in the primary system can drain by gravity into the reactor core, which is located at the lowest eleva tion in the plant. 2. Diesel generators, required for injection system pumps in current de signs, have been eliminated. 3. Cooling the core after a LOCA is accomplished by depressurizing the reactor and then allowing water to flow in by gravity from overhead tanks. 4. Passive systems are used to cool the reactor containment in the event of an accident. 5. Reactor power densities are reduced, thus increasing the margin of safety. 6. Plant designs have been substantially simplified. Some of the major PWR designs in this category have power levels in the range of 300 to 600 MWe and continue to use an external loop design with vertical stearn generators. Typical of this approach is the two-loop AP-600 reactor 31 designed to produce 600 MWe* (see Fig. 1 .1 5) . Note that in this design the reactor coolant pumps are of the canned motor design and are integral with the heat exchangers. A passive residual heat removal system removes core heat when feedwater fails. This is accomplished by a natural circulation loop, which exchanges heat between the primary circuit and an *A
l OOO-MWe version of this design (AP-I OOO) has also been proposed.
T H E RMAL A N A LY S I S OF PWRS
26
���9r---
D i s c h a. r g e
Air
Low
Pressure
\.I o. t e r
T o. n h:
E v o. p o r o. t i o n
C o o l ing Air
Inlet
Steel Pressurizer
I n - C o n t o. i n M e n t Refueling \.I o. t e r S t o r o. g e T o. n k < I R \.I S n
C o n t o. i n M e n i: V e ssel
Ai
B o. f f l e
Core
M o. h: e u p
T o. n h:
-r----t---+-_ A c e u 1"1 u l 0. t o r
Jf--===7�--L_ Fig. 1.15
R e a. c t o r
Vessel
Configuration o f advanced reactor (AP-600) with passive safety features (based on information provided by Westinghouse Electric Corp.).
in-containment refueling water storage tank. After a LOCA, water is first supplied by core makeup tanks maintained at system pressure and subse quently by two large accumulators pressurized to 49 bar. Long-term cooling after the system is depressurized is provided by an in-containment refuel ing water storage tank. This tank has a 1 0-hr supply of coolant. After this tank has emptied, the containment area will be flooded above the highest elevation of the reactor vessel. Long-term cooling is maintained with con densation of boil-off by the air-cooled vapor container walls. A more radical departure from present practice is those advanced con cepts that use "integrated" designs in which steam generators and pumps are within the reactor vessel. The primary loops are thus eliminated. An arrangement of this nature is proposed32 for the Safe Integral Reactor (SIR). In this system, shown in Fig. 1 .1 6, the core is at the bottom of the pressure vessel and placed under a tall riser to enhance natural circulation. A num ber of steam generators and sealed primary circulation pumps are located
POW E R G E N E RATION
./
/�
F LU I D I C
D I O D E �.
CONTROL
RODS
(65
TOTA L )
�- P R E S S U R I Z E R
S P RAY R EACT O R PU M P OF
(1
C O O LANT �--����
6)
F LU I D I C
DIODE
STEAM F E E DWAT E R
�
STEAM G E N E RATO R
� N AT U R A L C I R C U LATI O N F LU I D I C D I O D E S STEAM G E N E R ATO R
) ") l
Fig. 1 . 1 6
(1
OF
1 2)
! R EA C T O R CORE
Configuration o f safe integral reactor [from Ref. 29] .
around the periphery of the vessel. The upper part of the vessel contains a passive pressurizer system that regulates system pressure. In the SIR design, a large LOCA is considered impossible since there is no primary coolant piping. Potential seismic failure due to differential movement of the vessel and piping in current designs should be eliminated. A more detailed description of the AP-600 and other advanced designs is provided in Chapter 6. 1 .1 .3 . 3
PRIME Reactors
The reactors in this group propose radical departures from current PWR technology. The best known of these designs is the PIUS (Process Inherent
T H E RMAL A N AL Y S I S O F PWRS
28
Ultimate Safety reactor) originally proposed by Hannerz. 33 The reactor (shown in Fig. 1 .1 7) is a combination of a PWR and a swimming-pool re actor. The core is placed at the bottom of a very long standpipe or riser located in a large pool of water. Piping at the upper end of the standpipe conducts the circulating flow outside the reactor vessel to one of several steam generators and pumps. The water is then returned to the vessel and flows through an annular downcomer to the core. The lower end of the standpipe containing the core has an opening that connects it to the pool of borated water in the pool. Under operating conditions, the pump head keeps borated water from entering at the bottom of the standpipe. Although
pressurizer s t e a r'\ volur'\e
(J
o (J
siphon Ioreaker arranger'\ent
��"
Feed-water
Upper density lock
a
(J
Reactor core
o
o
l)
P r e - s tressed
o
concrete reactor vessel
o Lower den s i t y
o
o
Fig. 1 . 1 7
PIUS reactor
lock
o 17
configuration [based on information from
Nucl. News, p. 85
(Sep.
1992)].
POW E R G E N E RATI O N
29
the standpipe is also connected to the pool at the upper end, the pressure at the upp er end of the standpipe equals the pressure in the pool. Therefore, pool water does not enter the standpipe during normal operation. In the cas e of a loss-of-coolant flow, a natural-circulation loop would be set up with pool water entering at the bottom of the standpipe and exiting at the top . The core, riser, pressurizer, and water pool are all located within a large (12 to 13 m. i.d.) prestressed concrete pressure vessel. A double stainless steel liner protects the concrete from the borated pool water and prevents leakage. Natural circulation of the pool water through air-cooled heat ex changers provides enough heat removal to keep the pool water cooled un der both normal and accident conditions. In the event that air access to the heat exchangers is curtailed, there is enough pool water to keep the core covered and cool for at least one week after shutdown. The Japanese System Integrated Pressurized Water Reactor (SPWR) con cept is similar to PIUS in a number of respects. The SPWR has a core located at the bottom of a long riser in a large vessel. However, in this case the vessel is made of steel and is considerably smaller than the vessel proposed for PIUS. In the SPWR, both pumps and steam generators are placed within the reactor vessel. Advanced designs for CANDU-type reactors have also been proposed. 29 In these designs, the pressure tubes have the unique characteristic of be coming highly conductive to heat (instead of acting as an insulator) once a given temperature has been exceeded. The fuel decay heat can then be transmitted to the water in the low-pressure calandria. Heat pipes or other mechanisms then conduct this heat to the environment. 1 . 1 .4
Status of Various Concepts
Although a variety of pressurized water concepts for generation of central station power have been examined, many of these are no longer actively considered. For large central stations, only three design types were in op eration in 1 994: ( 1 ) the vessel-type heavy-water moderated design, (2) the CANDU heavy-water-moderated design, and (3) the vessel-type light water-moderated system at subcritical pressure (conventional PWR). The vessel-type PHWR design has been used for only two reactors in Argentina (1 994). The CANDU design is used extensively in Canada and is also used in Pakistan, India, Korea, and Argentina (worldwide total of 30 units in 1994). However, the light-water-moderated conventional design, using slightly enriched fuel, has dominated PWR plant construction in the United States, Western Europe, Japan, the former Soviet Union, and Eastern Eu rope. More than 240 plants were deployed worldwide in 1994. Since the conventional PWR dominates PWR practice, this monograph concentrates on its analysis.
T H E RMAL A N A L Y S I S OF PWRS
30
Some of the advanced concepts described in Section 1 . 1 .3 are likely to be important in the future. However, it is too early to determine which of these will be widely employed.
1 .2 1 .2.1
STEADY-STATE POWER G E N E RAT I O N A N D D I STRI B UTION I N CONV E N T I O N A L PWR CORES Power D istribution in Unpertu rbed, U niformly Loaded Cores
Since light-water-moderated reactors cannot go critical using entirely nat ural uranium fuel, the designers of the early PWRs had three choices:
1.
To utilize fuel elements that all contain slightly enriched fuel 2. To provide a on which highly enriched fuel elements are uniformly dispersed in a matrix of natural uranium elements 3. To provide a region of highly enriched fuel surrounded by low enrichment or natural uranium fuel.
spiked core,
The last alternative formed the basis of the seed-blanket design. Dispersion of enriched elements among natural uranium fuel results in power outputs for the enriched rods substantially greater than for the others. It is difficult to provide additional flow to the enriched elements and they could limit power output. Further, significantly lower fuel costs can be obtained by using slightly enriched fuel in all elements. This is the common practice for large, central station power plants. No practical reactor design has used a spiked core. If the fuel elements were dispersed through the core in a uniform manner and if fuel enrichment were uniform throughout, we can roughly approx imate overall neutron behavior by considering the core to be homogeneous. Most light water designs are essentially unreflected. Thus, we consider the unperturbed core as a bare homogeneous reactor. For such a core, an ap proximation of the neutron flux distribution can be obtained by solving the wave equation34 (1.1) where
i s thermal neutron flux and B 2 i s geometric buckling. Our bound ary conditions for the solution of this equation are that flux goes to zero at the extrapolated boundaries of the reactor and that it is finite everywhere except at the extrapolated boundaries. The extrapolated boundaries are ap proximated by adding 0.71 �tr t where �tr i s the transport mean-free-path (mfp) of a neutron, to the distance from the centerline. Thus, for a cylin drical reactor, extrapolated length is
L' L' L + 1 .42 � tr , =
0.2)
POW E R G E N E RATION
TA BLE 1 . 1 Thermal Flux D istribution in Bare H omogeneous Reactors
Distribution Function
Coordinate
Geometry
(��) (��) (��) (�n cos
Infinite slab Rectangular parallelepiped
x,
Y, or
cos
z
, cos
, cos
sinCrrr / R ) TIr / R I
Sphere
fo (2.405 r / R ' ) I
{:
Finite cylinder
cos
(�n
where L is the actual length of the reactor and R ' the extrapolated radius, is given by R =R+ '
0.71
A tr
0.3)
where R is the actual radius. For large reactors, A tr is negligible with respect to R and L and can be ignored. The solution for several of the most common geometries is listed in Table (Ref. Under the conditions we have postulated, and if the effects of burn up are not significant, power generation is proportional to thermal neutron flux. Thus, the functions indicated in Table provide an overall description of power distribution. Power distribution within a reactor free of control rods is not only of academic interest; two control systems lead to such a situation for base loaded plants. In the chemical shim control system, boric acid is added to the reactor to compensate for excess reactivity at the beginning of life (BOL). However, it is gradually withdrawn as core reactivity declines with lifetime. A similar procedure is used in spectral shift controt * where the ratio of D20 to H20 in the moderator is decreased in response to the change in reactivity with lifetime. Decreasing the amount of D20 present increases the number of thermal neutrons present in the core, thus increasing reac tivity and compensating for the decrease in reactivity due to fuel burnup. At full power, such reactors can be operated with all control rods essentially withdrawn.
1.1
35).
1.1
*The original spectral shift control system is no longer of commercial interest. However, similar approaches are used. The calandrias of some D20-moderated pressure tube reactors are pierced by vertical hollow tubes, which can be filled with H20. By varying the H20 level, the neutron spectrum can be varied. A similar idea is employed in some advanced vessel type PWRs, in which a portion of the control element clusters are fitted with water displace ment rods. Movement of these nonabsorbing rods changes the moderator-to-fuel ratio and thus varies the neutron spectrum.
T H E RMAL A N A L Y S I S O F PWR S
32
Power distribution at BOL, in the uniform, unrodded cylindrical reactor, is closely approximated by a radial distribution and an axial cosine dis tribution. Thus, we write
fo q (2.405r / R ' ) (��) q ":nax m
q qr
where
= = "�ax = z = III
=1 0
cos
(1.4)
rate of volumetric heat generation maximum volumetric heat generation at the center of the core radial loca tion axial location (from centerline).
1992
Most PWRs operating in utilize fuel asssemblies that are uniformly enriched in the axial direction. In the absence of part length burnable ab sorbers, a cosinusoidal flux and power distribution in the axial direction is thus a good approximation at BOL. As burnup proceeds, the flux and power distribution are flattened both axially and radially. This results from the more rapid depletion of burnable isotopes in the high-power regions. Since power peaking is generally reduced as the cycle proceeds, the limiting thermal-hydraulic conditions generally exist at the beginning of the cycle. 1 .2 . 2 1 .2 . 2 . 1
Effect of Fuel load ing Standard Core Arrangement
Most of the early power PWRs were uniformly loaded. In very small plants, where simplicity of refueling is important, this loading scheme must still be useful. For large, central station plants, one disadvantage is that it leads to high power peaking in the central region of the core. In addition, rela tively low-average burnups are obtained since even at the end of life (EOL) the outermost fuel elements have been only moderately burned down. One means of alleviating these difficulties is to use a zone loading scheme, where the core is loaded with fuel elements of three or more different en richments. The most highly enriched fuel elements are placed in the outer most position; the least enriched are placed in the central region. Since power is approximately proportional to the product of thermal neutron flux and fissionable fuel concentration, the power level in the inner region of the core is lowered and that of the outer region is raised. This power flat tening can significantly increase the total power capability of the core. When criticality is no longer possible, the entire core is replaced. Replacement of the complete core at each refueling leads to a low average core burnup. This is increased in most fuel cycling schemes by replacing only a portion of the fuel at each refueling. In the simplest of these schemes, the initial core is zone loaded as described above. At the
POWE R G E N E RATI O N
BATCH LOA D I N G
T H R E E-REG I ON C H E C K E R BOAR D
T H R E E-REG I O N
� • o II
Fig. 1 . 1 8
Fuel loading patterns (based on information from Westinghouse Electric Corp. Fuel Technology Update No. 1 1 ) .
end of the first core life, the most burned fuel from the central core region is removed, the outer regions are moved inward, and fresh fuel is placed in the vacated spaces in the outermost region. The same procedure is fol lowed in all subsequent core loading so that at equilibrium all the removed fuel has been through three or more cycles and has a high burnup. Batch and three-region zone loading are illustrated in Fig. 1 .1 8. Several variations of this fuel cycling scheme are available. An early im provement was the so-called "Roundelay" method. In the equilibrium cy cle, fresh elements are inserted in uniform distribution throughout the core at each refueling. For example, with Roundelay batch, every third element is replaced while the other elements are left in place. In Fig. 1 . ] the fuel elements are numbered in the order in which they might be replaced in the reference core; i.e., during one refueling all No. 1 fuel assemblies would be replaced, and so on. Mixing fresh and burned assemblies produces a strong Coupling between them and permits improved power production from the burned elements.
3
9,
T H E RMAL A N A L Y S I S OF PWR S
34
Vesse l 202-i n. i . d
3
3
3
1
1
3
3
1
2
l- 8.361 i n .
2
2 1
--J
1
2
2 3
2
2 3
1
2
3
1
1
2
3
2
3
1
1
2
3
3
1
2
3
2
1 50.458-i n . Core Wi dth
1
3
2
3
1
2
3
1 3
2
1
3
-J
1 50.498-in. Inside Baffl e Wi dth
Fig. 1 . 1 9
Roundelay fuel cycling scheme (from Ref. 36).
1.20
Figure shows a typical radial power distribution in equilibrium cores after refueling for three-region and Roundelay-loaded reactors. The power distributions shown are the ratio of the power at the given point to the average power across the core at that axial location. The marked "rip ple" in the Roundelay power distribution results from significantly differ ent power production from adjacent fuel assemblies with different burnups. In general, for both refueling schemes, as burnup proceeds, peak power decreases from that shown and power in the lower power region increases. Note that small changes in fuel assembly placement can significantly affect peak power. The three-region checkerboard scheme illustrated in Fig. was some what more widely used than the Roundelay approach. In this refueling procedure, the outer region is fresh fuel, but the innermost region is a checkerboard pattern of second- and third-cycle fuel. The actual determi nation of the location of the specific assembly locations in the central region is fairly sophisticated. The final arrangement resulting from these studies may not strictly conform to the checkerboard pattern shown in Fig. This is certain to be the case if fuel failures require the insertion of extra
1. 1 8
1. 1 8.
35
POWE R G E N E RATION
. 1> 1
. 312
. 31 1
. 210
( I . �) ( 1 . 282) (1. 322) ( 1 . 512) . 777
. 554
. 691
. 441
( 1 . 1 43) ( 1 . 167) ( 1 . 1 58) ( 1 . 303) . �8 . 891 . 661 . fJ79 1 . 021 .522 ( 1 . 1061 ( 1 . 0971 ( 1 . 13 71 ( 1 . 1621 ( 1 . 286) (1. 4921 . 549 . 746 1.012 1 . 088 1 . 029 1 . 1 98 (1. 0701 1 1. 1 1 1 1 ( 1 . 082) (1. 130) ( 1. 1371 ( 1 . 287 ) 1 . 066 . 745 1 . 299 1 . 150 1 . 490 1 . 315 ( 1 . 074) ( 1 . 084) ( 1 . 080) ( 1 . 0751 ( 1. 129) U. 1571 1 . 368 ( 1 . 1011 1 . 463
1. 583 1 . 396 1 . 202 1 . 329 ( 1 . 075) ( 1 . 07 4) ( 1 . 07 31 U . 0 7 5) 1 . 641
1. 394
1 . 424
(I. Q.:¥)) ( 1 . 1271 ( 1 . 0751 U. 0751 1. 836 1 . 622 1 . 438 ( 1 . 06 1 1 ( 1 . 0661 ( k f2 1 1 1 . 604
I,:�
( 1. 126) U.
I.
�
�. 1
4)
-
r-
FR
•
�
1 . 99 -
23)
- -
FOR EACH A SSEMBLY Upper power Lower power Fig. 1 .20
number i s (average power i n assem bly}/(average in core). number is (maxi m um power in assem b l y}/(average in assembl y) .
Typical radial power distribution i n Roundelay-loaded reactor (from Ref. 36).
scatter-loading
fresh fuel or the reuse of the third-cycle fuel. A detailed study of possible scheme, which provides the placement schemes is made; the best economics, is selected. In view of the very large number of placement policies that can be followed, a methodical procedure limiting the inter changes considered is required. Rothleder37a lists a series of heuristic rules that can be used to limit the number of policies considered. More recently, low leakage fuel loading patterns, such as the pattern indicated by L3p in Fig. 1 . 18, have been introduced. In these fuel patterns, fresh fuel is no longer in the peripheral positions. This has the advantage of reducing neutron leakage from the core and allowing fresh fuel of a lower enrichment to be used. In addition, the lower neutron flux at the core edges means lower rates of reactor vessel embrittlement and hence leads to lo nger reactor vessel lifetimes. Ultra-low-leakage fuel pattern schemes such as that indicated by L4p in Fig . 1 .18 have also been used. Such schemes provide additional improve-
TH E RMAL A NA L Y S I S O F PWR S
36
ments in neutron economy and some additional reduction in the fast neu tron flux at the reactor vessel wall. If the presence of impurities such as copper have reduced the ductility of reactor vessel welds, it may be desir able to further reduce the flux at the vessel wall by insertion of absorber rods (e.g., stainless steel or hafnium) in the assemblies closest to the welds. Since power is shifted from the core periphery to interior locations by low-leakage fuel management, higher peaking resul ts. Acceptable peaking factors are obtained by insertion of burnable poison rods in those fresh assemblies with high peak powers (Fig. Burnable poison materials, such as those containing gadolinium, erbium, or boron, provide a significant reduction in reactivity at BOL. As the cycle progresses, the poison is substantially depleted by interaction with the ther mal neutrons present. At the end of the first cycle, the poison is ahnost entirely consumed and hence it causes only a slight decrease in end-of cycle reactivity. If the poison were contained in rods placed in the rod cluster control (RCC) tube, the poison rods would be removed from the assembly at the end of the first cycle. When the poison is mixed with fuel, removal at the end of the first cycle is, of course, not possible. Because addition of burnable poison significantly reduces beginning-of cycle (BOC) reactivity, less boric acid must be dissolved in the moderator
1.2 1).
1 .4 >< :::l
I
I
...... -
c
o J-t
-
:::l Q) C Q)
::: B
.
......
Q) 0::
0.8 0.6 0.4
0.2
0.4
�a
0.6
0.8
Frac t i o n al d i stance t ro m c o re c e n te r Fig. 1 .2 1
Effect o f burnable absorbers on radial flux flattening (based o n information provided by Westinghouse Electric Corp.).
---�---L�--�
POW E R G E N E RATI O N
to produce a multiplication factor of 1 .0. This is highly desirable when long cycle lengths (i.e., 18 or 24 months) are used. If dissolved boric acid alone were used to reduce BOC reactivity, the high boron concentration required is likely to lead to a positive moderator temperature coefficient. Since a negative moderator temperature coefficient is required by U.s. Nuclear Regulatory Commission (NRC) regulations, long refueling cycles are not feasible without the addition of burnable poison. Although all PWR fuel designed prior to about 1990 had an axially uni form enrichment, some later designs have included axial blankets.37b These blankets, which are about 6 in. long and placed at both ends of the fuel assembly, contain pellets produced from natural uranium. The axial blan kets serve the same purpose as the low-leakage x-y positioning of the fuel assemblies, reducing neutron leakage. The enrichment of the center portion of the fuel stack is increased to compensate for the natural fuel at the ends. However, the average enrichment of the entire assembly is reduced slightly resulting in an overall savings of 1 to 2% in fuel cycle costs. Since the power production in the end regions is reduced, the power in the central region is correspondingly increased. Without burnable absorbers, this can result in a 6 to 8% increase in axial power peaking. This peaking can be counter acted by burnable absorber rods. When axial blankets are present, the neu tron flux and power drop sharply in the blanket region. A cosine function no longer provides an adequate description of the axial flux variation. 1 .2 .2 .2
Use of Mixed-Oxide (MOX) Fuel
The availability of reprocessed plutonium in Europe has led to the use of plutonium-fueled (U0 enriched with PU02 ) assemblies in some PWRs. As 2 of 1 992, when fuel containing plutonium has been used, it has made up only a portion of the fresh fuel added to the core (e.g., 30%).38 Since the nuclear characteristics of plutonium and 235U differ significantly, the power peaking in the region of a plutonium-enriched assembly adjacent to a 235U_ enriched assembly poses a problem. Both the fission and absorption cross sections of 239pU are higher than those for 235U. Therefore, to obtain the same reactivity worth of a fresh assembly, the Pu assemblies have a higher average enrichment than those with 235U. Since the flux must be continuous across an assembly boundary, the higher enrichment would produce an unacceptable high heat flux if it were used in the fuel rods on the periphery of the assembly. This difficulty can be avoided by using three concentric rings with three different enrichments in the same assembly. The outermost zone contains fuel with the lowest enrichment, whereas the innermost zone contains fuel with the highest enrichment. A typical arrangement would have fuel with approximately 4% Pu in the outer zone, 6% in the middle zone, and 8% in the innermost region. An alternative way of dealing with Pu-enriched fuel assemblies is to use reconstitutable assemblies and use three burnup regions within the assem-
T H E RMAL A N A L Y S I S OF PWRS
38
bly?8 A kind of in /out reloading scheme would be used. At equilibrium, the following rearrangement would occur at the end of a cycle: 1 . Thrice-burned fuel pins of the outermost zone are unloaded and re moved from the core. 2. Twice-burned fuel pins are unloaded from the middle zone and loaded into the outer zone. 3. Once-burned fuel frOlTI the central zone is moved to the outer zone. Fresh fuel is added to the central zone.
4.
1 .2 . 2 . 3
High-Con version-Ra tio Core Arrangements
seed-alld-blanket
The earliest efforts to obtain a core with a high conversion ratio were based on the design of the Shippingport reactor. This reactor was the first PWR to be built for civilian power production. A seed-and-blanket core is an alternative to a design using slightly en riched uranium throughout the core. Such a core consists of one or more relatively small seed regions containing highly enriched fuel assemblies, and blanket regions of natural or low-enrichment fuel surrounding the seed regions. The blanket is composed of material with a multiplication factor less than unity and cannot be made critical by itself. Reactivity of the core is primarily determined by the properties of the seed; thus, control rods are required only in the seed. The geometry initially considered was that of a cylindrical annular seed with blanket regions both inside and outside. This arrangement (see Fig. 1 .22 and Refs. 39 and was used in Shippingport core 1 . The blanket elements were bundles of zirconium-alloy-clad cylindrical rods containing U02 pellets fabricated from natural uranium. The seed elements consisted of a series of fuel plates held by Zircaloy alloy boxes, each segmented into four quadrants. The space between the quadrants was used for a control rod passage. The fuel plates themselves consisted of an enriched fuel alloy strip sandwiched between two zirconium alloy cover plates and four side strips. In later seeds, a dispersion of U02 in Zircaloy replaced the U-Zr alloy fuel. The power distribution within a seed-and-blanket reactor is appreciably different from the distribution heretofore considered. In reactors of this type, infinite multiplication of the annular seed region is greater than unity, whereas that of the blanket is less than one. The higher fuel content and multiplication factor of the seed result in a higher fast neutron flux in the seed . In addition, the highly enriched fuel in the seed has a higher thermal absorption cross section than the blanket. This results in a leakage of fast neutrons from the seed into the blanket and a leakage of thermal neutrons in the opposite direction. In the blanket, 238U fast fission produces the power at BOL. As burnup proceeds, formation of plutonium in the blanket creates new power-
40)
39
POW E R G E N E RATI O N
Pressure Vesse l Carbon Ste e l w i th O . 2S - i n . Sta i n l ess- Stee l C l add i ng ----,
Core Shroud Therma l Shi e l ds B l anket Reg i on 1 2 1 Element s B lanket Reg ion 2 24 Elements Seed Reg ion 32 E l emen ts B lanket Region 3 40 Elements B lanket Reg ion 4 28 Elements
C
I 0o
Inl et Nozz l e Fig. 1.22
Cross section o f Shippingport reactor core [from Directory of Nuclear Reactors, Vol. IV Power Reactors, pp. 21-32 (1962)].
producing fuel. Leakage of fast neutrons from the seed thus results in power production in a region that cannot by itself maintain a chain reaction. It is economically desirable for as much power as possible to be produced in the relatively low-cost natural uranium blanket. The power division between blanket and seed is one of the important fa ctors in determining the thermal design of a reactor of this type. Radkowsky and Bayard41 discuss the procedures used for calculating the power distribution in Shippingport core 1 . At BOL, the ratio of blanket power to total power was calculated as Seed power does not decrease as rapidly as seed volume and, therefore, a decrease in seed volume results in increasing seed power density. Hence, power sharing tends to be determined by the maximum power density that can be extracted from the seed. Thus, the average power density in the seed can be more than four times that of the blanket.4 1 Since resonance capture
�O.52.
T H E RMAL A NALYS I S O F PWR S
40
Out l e t N ozz l e
Mova b l e Seed
Re f i e ctor --+'7-'?+---R---fII'" Bla nket
S ta t i onary B l anket
Fig. 1 .2 3
Cross-sectional view o f the first Shipping port
LWBR
core.
is of no consequence in the seed, it is feasible to use thin fuel elements with a high surface area per core volume to attain the high power removal rates required. Inlet orifices can be used to provide considerably higher coolant mass fluxes to the seed than to the blanket. Later studies with the seed and blanket concept were directed toward developing a light water breeder reactor (LWBR). The concept was based on the 232 Th_233 U cycle. Both the seed and blanket consisted of Th02 fuel rods enriched with varying amounts of 233 U02 . (Seed rods contain up to 6 wt% U02 ; blanket rods contain up to 3 wt% U02 .) From the cross section of the first Shippingport LWBR reactor (Fig. 1 .23),42 we see that the core consisted of a symmetrical array of hexagonal modules surrounded by a reflector blanket. Each core module contained an axially movable seed re gion 1 ) and a stationary annular blanket Both seed and blanket regions were made up of arrays of tightly packed cylindrical fuel rods con taining oxide pellets. Control of the LWBR was accomplished by moving the seed. As the seed elements were moved out of the core, leakage was increased and the core
(k>
(k
POWE R G E N E RATI O N
became subcritical. This control procedure, which avoids losing neutrons to poisons, was necessary since the calculated breeding ratio was only slightly greater than unity. The results obtained from operation of the LWBR appear to indicate that, when all reprocessing and fabrication losses are considered, true breeding is probably not possible. This conclusion, and the obvious complexity of seed-and-blanket cores with movable seeds, has led to the virtual abandon ment of this approach. While the seed-and-blanket approach continues to have an occasional advocate,43 most recent interest has been focused on homogeneous tight lattice cores4�6 using the uranium-plutonium cycle. In this approach, orig inally suggested by Edlund,47 the standard PWR core is replaced with a core formed by hexagonal fuel assemblies containing fuel rods on a trian gular pitch. By using a tight lattice, the water volume fraction is reduced, thus leading to a harder neutron spectrum. When such a lattice is fueled with plutonium discharged from LWRs, the hard spectrum and the signif icant percentage of 241 pU in the fuel allows conversion ratios of to to be achieved. This compares with conversion ratios of to found in conventional PWRs. With a conversion ratio of approximately 4% of the uranium fuel can be utilized in contrast with a 1 % utilization when Pu is recycled in a conventional LWR. These reactors are homogeneous in the sense that all fuel rods have iden tical shapes. However, the plutonium concentration may vary with position in order to reduce power peaking. Both axial and radial blankets would be used. In most designs, the outer row of assemblies serves as the radial blanket and these assemblies contain no Pu at the beginning of life. Both Zircaloy and stainless steel44 have been proposed for cladding ma terials. Because of the epithermal neutron spectrum, stainless steel produces much less of a poisoning effect than it would in a conventional PWR. Fur ther, the higher strength of stainless steel results in considerably less dis tortion if the fuel is exposed to the high temperatures possible during a LOCA. This is important in maintaining a coolable geometry with the small clearances available in a tight lattice. Another advantage to the use of stain less steel is that it can be formed with integral spacer ribs (Fig. 1 .24). These avoid the difficulty of fabricating spacer grids for tight lattices and, in ad dition, such spiral ribs improve coolant heat transfer characteristics. Table 1 .11 comp ares the geometrical characteristics of a proposed tight lattice de sign to those of conventional PWR designs.
0.8 0.9 0. 5 0. 6 0.9,
1 .2.3 1 .2.3. 1
Effect of Control Rods, Water Slots, Voids, and B u rnable Absorbers Effects of Control Rods on Power Distribution
In some reactor designs, all excess reactivity is compensated for by inserting control rods. The presence of the strongly absorbing material perturbs the flux both axially and radially. However, this is not entirely a disadvantage
T H E RMAL ANALYSIS OF PWR S
42
Fig.
1 .24
Fuel rods with integral spiral spacer ribs for tight-lattice core [Broeders and DaIle-Donne, Nucl. Technol., 71, 85 (1985) ] .
TAB L E 1 . 1 1 Geometrical Characteristics o f PW R Fuel Assembl ies
Reactor Type Standard PWR, u.s. or West European design Standard PWR, Russian design Tight-lattice PWR
Lattice Type
Typical Fuel Rod Diameter (mm)
Typical Minimum Clearance Between Fuel Rods (mm)
Fuel Rods per Assembly
Fuel Rod Length (cm)
Square
9.5-1 0.8
3.5
204-300
330-440
Triangular
9.1
3.1
1 26
255
Triangular
9.5
1 .9
313
220
43
POW E R G E N ERAT I O N
Control Rods
---/
Power Distribution i n Unperturbed Core
Power Oi stri bution w i th Centra l Rods Inserted
Co re Radi us Fig.
1.25
�
Radial power distribution in a cylindrical reactor with and without control rods.
since the rods can be used to improve radial flux distribution. For example, some, or all, of the rods in the center of the core can be partially inserted at BOL, thereby reducing flux peaking. This effect is illustrated in Fig. 1 .25, where we observe that flux essentially goes to zero at the boundaries of the control rods, as well as at the extrapolated boundary of the reactor. The flux and, hence, the power level in the outer region are increased concur rently with the decrease in the central region. The improvement in radial distribution attainable by inserting rods is often more than negated by the change in axial distribution. The usual situation in a PWR where the rods are top entering is shown in Fig. 1 .26. At BOL, the partially inserted rods skew the flux toward the bottom. As the rods are withdrawn, the less burned fuel at the top of the core causes the flux to be skewed toward the top at EOL. As observed in Fig. 1 .26, the ratio peak-to-average power can be higher than that of the unperturbed core. Power distribution of the type shown in Fig. 1 .26 generally can be rep resented by modifying the usual cosine distribution to the form = (A
(l) + Bl )
+1
cos(ao l )
(l .5)
where A, B, and ao are constants and I is a dimensionless length coordinate va rying from - 1 to along the heated length of the core. Expressions of this form are applicable only if the ratio peak-to-average flux is < 1 .892. Establi shing constants and using expressions of this form are discussed in Sec. 4.1 . 1 .
TH ERMAL ANALYS I S OF PWR S
44
Uncontro l l ed ( Idea l i zed) -
- -
-
-
-
-
-
- - ;0,---__
q"
(a)
(b ) ..
L
End-Of-li fe
(c) ..
Fig.
1 .26
L
Axial power distribution.
Chemical shim control, via addition of boric acid to the coolant, has been almost universally adopted for currently operating (1994) conventional PWRs designed for central power. This means that, if the plant is base loaded, these reactors operate with control rods almost entirely withdrawn at full power throughout core life. Further, axial flux distortion due to rod motion is minimized. Note that addition of boric acid is also used to com pensate for the reactivity increase in going from hot-zero power conditions to cold-zero power conditions.
POW E R G E N ERAT I O N
When the shutdown rods are maintained in exactly the same all-rods out (ARO) position throughout core life, excessive wear can occur in the upper internals area where the guide tubes interface with the rod lets of the control elements. To avoid this problem, it is desirable to reposition the rods slightly during the cycle. It has been suggested48 that at the beginning of the cycle the control elements be inserted to a position slightly below the top of the outer core. The rods are then periodically withdrawn, one step at a time, so that at the end of the cycle they are slightly above the active core. This approach avoids undesirable axial flux peaking as well as maximizing EOC shutdown margin. Although base load operation of nuclear power plants is possible in power systems where most of the power generation is non-nuclear, a por tion of the nuclear plants needs to be load following when a substantial fraction of the system generation is nuclear. Some power variations will always have to be accommodated even in plants that are primarily base loaded. The procedures used to minimize flux distortion following a load change are discussed in Sec. In some advanced core designs, only some of the control rods are ab sorber rods. Most of the rods used are water displacement rods of Zr02 encapsulated in a zirconium alloy. These water displacement rods are fully inserted at the beginning of the cycle and then sequentially removed as the cycle progresses. Much of the change in reactivity with lifetime is thus compensated for by these rods through a shift in the neutron energy spec trum. The conversion ratio is improved because of the reduced amount of boric acid needed in the coolant and because of the harder neutron spec trum. The combination of burnable poison added to some fuel rods plus water displacement rods can allow the complete elimination of chemical shim control. Advanced cores with water displacement rods require that all the RCC guide tubes be used rather than only the tubes directly under a drive mech anism. Control rod clusters are redesigned so that one spider controls water displacement rods in four assemblies. To limit the number of rods in a cluster, the number of guide tubes per assembly is reduced, but each guide tube replaces four fuel rods instead of one.
1.5.
Control Rod Beha vior. In those load-follow designs in which some ab sorber rods are inserted for an appreciable fraction of the operating time, the behavior of the absorber material at high neutron fluences is of concern. Experience has shown that the Ag-In-Cd alloy (80 wt% Ag, wt% In, wt% Cd) used in many designs has good irradiation stability. Since the corrosion resistance of the alloy in high-temperature water is unsatisfactory, the alloy must be encapsulated in stainless steel or Inconel. Rods using this material have performed well for more than 10 years. Although somewhat less desirable than Ag-In-Cd, boron carbide absorb ers have often been used because of their lower cost. In addition to deple-
15
5
46
TH E RMAL ANALYS I S O F PWRS
tion of the boron, long-term operation leads to helium production and swelling of the B C. These phenomena can stress the cladding and lead to 4 cladding failures. Although such failures have been a problem in BWRs where the rods receive neutron fluences generally above those seen by PWR rods, PWR operation appears to be satisfactory. Hafnium is an excellent absorber and has both good corrosion resistance and dimensional stability. Although it has been used in naval reactors and in the Shippingport reactor, it has not been used in subsequent central sta tion PWR power plants. 1 .2.3.2
Power Perturbations D u e to Structural Ma terial, Wa ter Slots, and Voids
No fuel-water lattice is entirely uniform; additional structural material is present at all grid locations, in fuel assembly cans, in the support structure required at core edges, etc. Since such material is a neutron absorber with out being a moderator or source, it causes a local decrease in flux and, hence, in power. The effect can be small if the material is of low cross section (i.e., Zircaloy), but it can be substantial if a high cross-section ma terial, such as stainless steel, is used. Perhaps of even more concern to the designer is local peaking due to additional water slots in a light-water-moderated reactor. Such spaces result from clearance between fuel assemblies, variations in lattice dimensions, and slots left by the withdrawal of control rods. Since water provides ad ditional moderation, the local thermal flux and, hence, power, increases. This effect is illustrated in Fig. 1 .27 for a circular water hole in a low enrichment, stainless-steel-clad core. 49 Slots of a size that could cause sub stantially more peaking than illustrated can easily arise; care is taken to avoid these. For example, where cruciform or Y -shaped control rods are used, followers of a low cross-section material are provided to prevent large water slots from being introduced when the rods are withdrawn. Such fol lowers are not required with RCC and, as previously noted, the RCC con cept allows the use of shorter reactor vessels. Creation of a steam void at the exit of the warmest core region is an additional factor that distorts normal flux patterns. In general, replacement of water moderator by steam reduces core reactivity and, hence, flux and power, in the region of the voids. The effect of such voids is particularly significant during transient and accident situations where coolant enthalpy rises appreciably above its normal value. Power reduction due to void cre ation can decrease the severity of some situations. 1 .2 .3 . 3
Burnable Absorbers
We have noted previously that the addition of burnable poisons or absorber rods is useful in (1) suppressing BOC power peaking and (2) allowing a negative moderator temperature coefficient to be maintained by reducing
47
POWE R G E N E RATI O N
Q)
0 :c
'0
Q) m ""U LU
"0
Q)
U
11'1
0 Q)
1 .10
.E
)( :l
u:
1 .0 0.0
0.2
0.4 Diame te r of Wa t e r Ho l e , i n .
Fig.
1 .2 7
Flux peaking at the edge of a circular water hole [from Nucl. Eng. Des.,
6,
30 1 (1967)].
BOC soluble boron concentrations. Originally, the burnable absorber was added to the core by placing burnable poison rods in some of the vacant RCC positions. The first of these rods consisted of an annulus of borosilicate (Pyrex) glass enclosed in a stainless-steel tube. Subsequently, some reactors used the wet annular burnable absorber (W ABA) design. The design con sists of annular pellets of a mixture of aluminum oxide and boron carbide (3.5 to 7 vol% B C) contained within two concentric Zircaloy tubes. Since 4 coolant (moderator) flows through the central tube, the additional neutron moderation in the central hole reduces the flux variation across the ab sorber. This provides a more even burnup of the boron poison and less of a reactivity penalty at the end of the cycle. Solid pellets of aluminum oxide and boron carbide have also been used as burnable poison rods. These poison rods are removed from the fuel assembly at the end of the first cycle. The burnable absorber rods will, of course, reduce the flux in the fuel assemblies in which they are inserted. In addition to this assembly-wide flux reduction, there is a significant local flux dip in the vicinity of each of the absorber rods. A further disadvantage is that burnable poison rods can not be placed in any assembly directly under a control rod drive position. More recently, burnable absorber material has been added directly to so me of the fuel rods. The absorbers chosen have been gadolinium oxide (Gd20 ) or erbium oxide (Er20 ) mixed directly with the U02 and formed 3 3 into pellets. Burnable absorbers of this type have been used to suppress
T H E RMAL ANALYSIS OF PWRS
48
corner rod peaking in BWRs for some time. The number of fuel rods con taining burnable poison can be considerably greater than the number of RCC positions in a fuel assembly and, therefore, lower poison concentra tions may be used. This leads to less of a flux dip in the vicinity of the absorber. Because lower absorber concentrations can be used, a more even radial burnout of the absorber can be achieved than with absorber rods placed in RCC thimbles. Assemblies with the burnable poison in the fuel rods can be located at any position in the core, thus providing greater flex ibility in refueling sequences. The poison rods remain with the assembly throughout its life in the core. An alternative to mixing the poison material intimately with the fuel is the use of a thin coating on the outside of the pellets. These integral fuel absorbers have used zirconium diboride (ZrB2 ) as the poison. Since the poison is on the outside of the pellet, it burns out uniformly. Properly designed absorber rods of this type lead to a very low residual reactivity penalty. Absorber rods can also provide some improvement in axial peaking fac tors. By eliminating absorber material from the regions at either ends of the assembly, some axial flux flattening is achieved . In any assembly containing an axial blanket, absorber material is always confined to a region that is somewhat shorter than the enriched fuel length. 1 .2.4
Nuclear H ot-Channel Factors
The thermal designer can readily determine average core parameters from a knowledge of total heat output, core heat transfer data, and flow to the core. However, core performance is not limited by average conditions, but by the most severe conditions. It is convenient and useful to define the hot channel of the core as that coolant channel or flow path where core heat flux and enthalpy rise is a maximum. Conditions in the hot channel are defined by several ratios of local-to-average conditions. These ratios are referred to as Those aspects of the complex nuclear data that are most significant to the thermal designer can be conveyed by three nuclear hot-channel factors. We define
hot-channel factors.
F�
= Radial
F� =
nuclear factor
Mean heat flux in hot channel mean heat flux in average channel of core
(1.6)
Axial nuclear factor Maximum heat flux in hot channel Mean heat flux in hot channel
(1 .7)
49
POW E R G E N E R A T I O N
F� = Nuclear heat flux factor Maximum heat flux in the core Mean heat flux in the core From the above definItions, it immediately follows that
0 .8)
FN_FNFN - Z R
0 .9) Some authors define as the highest value that the ratio (maximum chan nel flux-to-average flux of given channel) can have and, when so defined, the maximum of the ratio may not occur in the hot channel.* When using reported values of we should determine the basis for their definition. The above nuclear factors are defined on the basis of nominal channel dimensions. In computing limiting heat fluxes and enthalpy rises, consid eration must also be given to deviation from nominal dimensions and flow. The procedure used to account for these deviations is discussed in Sec.5.1 .4. An estimate of the magnitude of the nuclear hot-channel factors can be obtained by considering a large, bare, homogeneous reactor of cylindrical shape. For a uniformly enriched, unpoisoned, unperturbed reactor, we see that the heat flux follows a fa radial distribution and a cosine axial distri bution. Figure 1 .28 shows axial and radial nuclear hot-channel factors for such a reactor as a function of the ratios of (actual length) (actual length + extrapolation length). For large reactors of conventional design, the ex trapolation length becomes neligible, and the values for and D D ' approach unity. Thus, we find that under these idealized conditions F� = 1 .5708 X 2.3161 = 3.638. In an actual uniformly loaded core, the overall hot-channel factor would be higher due to such effects as lattice nonuni formities, control rods, changes in fuel and fission product concentration, and coolant property variation. These effects can often be more than com pensated for by improvements in that can be obtained through non uniform loading. Although the extrapolation length is negligible in most large cores, its effect should be considered in compact military cores. In some advanced core designs, a reflector or Zr02 encapsulated in zirconium alloy tubes is placed between the core baffle and core barrel. As will be noted in Section 1 .3, such a reflector effectively increases the distance beyond the core edge at which the flux extrapolates to zero. The effective for the core is then below one, resulting in a somewhat flatter radial power distribution. For a seed-and-blanket core, the overall core nuclear hot-channel factor has little meaning in view of markedly different power generations in the
F� F�,
Q
/ L / L'
/
F�
(L/ L ' )
*If it is then desired that the hot channel contain both the highest heat flux and enthalpy rise, it will not correspond to a real channel. In that case it should be considered as a hypothetical channel subject to the most severe thermal-hydraulic conditions possible.
T H E RMAL A N A L Y S I S OF PWRS
50
MPia. C O S ( Jr Z) 2 L
L'
L
M -.J o ( 2 . 4 0 4 8
L/L' 0,8 0,9 1.0 Fig. 1.28
I1x
1.3213 1.4313 1.5708
DID' 0,8 0,9 1.0
-b'
�
1.6571 1.9297 2,3161
Axial and radial nuclear hot-channel factors for unperturbed cylindrical reactors.
seed and blanket. It is useful to extend the concept to define hot-channel factors for each region of the core. We can define separate seed-and-blanket hot-channel factors by using Eqs. (1 .6), ( 1 .7), and ( 1 .8) with the word replaced by or The entire radial power distribution in any core can be described by defining a local radial factor, FJ1/, such that
seed blanket.
FN
Rl
=
mean heat flux in given channel Mean heat flux in average channel of core
core
(1.1 0)
For a typical core, Fig. 1 .29 illustrates the relative number of fuel elements with radial nuclear hot-channel factors in a given range. The peak radial flux is reached in a very small number of elements (often just one); the majority of the elements have radial factors < 1 .0.
POW E R G E N ERAT I O N
"'1J
o
0:::
Q) :::> u..
.....
o
... Q) ...0
E
:::>
Z
"0 Q) >
Q) 0:::
0.7
0.5
0.9
1 .0
1 .2
1 .4
F� Rad i a l N u c l ear Hot-C ha nne l Fa ctor Fig.
1 .29
1 6 ..
Power census in a typical rodded core.
An accurate value of F[5 for a real core requires a three-dimensional nuclear calculation for evaluating power throughout the core. The nuclear heat flux factor can then be calculated from FN Q
(l / V)
where
p ' (xo , Yo , zo )
ff f
0. 1 1)
P ' (x,y,z) dx dy dz
P ' (X,y,z) = local power density at (x,y,z) (xo , yo , zo ) = location of peak local power density V
:-:
core volulne.
Accurately determining the highest possible F [5 that could occur during life would require three-dimensional power calculations to be conducted for a very large number of conditions in a load-following plant. To avoid the expense of these calculations, a simpler synthesis procedure is often used. The synthesis approach defines two additional factors: Fxy (z) = Peak-to-average power density at axial elevation z P ' (z) = Relative core power at axial elevation
[ f:P ' (z) dz 1 ] =
T H E RMAL A N A L Y S I S OF PWRS
With these definitions, the value of F[1 is then given by 0 . 1 2) F[1 = max[ P ' (z)·Fxy (z)] In the synthesis approach, the values of p ' (z) are obtained from separate calculations. The values of p ' (z) are estimated from a one-dimensional axial calculation; values of Fxy (z) are obtained by two-dimensional calculations at a number of planes. By appropriate modeling, it is possible to obtain
1 . A one-dimensional axial distribution model that is consistent with three-dimensional average axial power distribution behavior during reactor operation 2. Two-dimensional values for Fxy (z) that are always equal to or greater than the value of Fxy (z) obtained for a three-dimensional calculation.5o The synthesis approach yields conservative values for F[1 under these con ditions. However, the more recent availability of rapidly running three dimensional nodal codes (see Sec. 1 .3. 1 .2) makes it possible to evaluate F[1 directly from a number of three-dimensional calculations if desired. In addition to using the synthesis approach to define an overall F[1, we can also define a heat flux factor at each elevation. That is, Ff5 (z) = [ P I (z) Fxy (z)] 0 . 1 3)
The values of Ff5 (z) and Fxy (z) obtained at the various planes examined may be connected to provide continuous Ff5 (z) and Fxy (z) functions. These curves then allow the calculation of the maximum linear power versus a function of elevation at full power reactor operation. The values so obtained are compared against the limiting curve of linear power versus elevation determined on the basis of LOCA and critical heat flux (CHF) considera tions. If the computed linear powers exceed the allowable, then reactor operation must be restricted so that the control rod arrangements leading to the unacceptable peaking are never used. 1 .2 .5
Heat Generation Within Fuel Elements
It is often assumed that the rate of power generation is essentially constant across the fuel rod. While this is often satisfactory as a first approximation, it is not strictly true. As previously noted, heat generation in a volume element of fuel of a thermal reactor is essentially proportional to the thermal neutron flux at that point. Since thermal neutron flux decreases toward the center of the element due to neutron absorption in the fuel, power gener ation decreases similarly. Following the explanation in Ref. 51 , we consider a unit cell of a uniform rectangular lattice containing cylindrical fuel rods of radius Ro . The prob lem is simplified by supposing that the square unit cell is replaced by a circle, radius R l ' with the equivalent area (Fig. 1 .30). Further, we assume
POW E R G E N E RAT I O N
Fig.
1 .3 0
Equivalent cell of fuel rod lattice.
that the neutron slowing down is constant in the moderator and zero in the fuel. The thermal diffusion equation for monoenergetic neutrons in the fuel rod is then 0 .14) where is neutron flux at any point in the rod, 0 is the diffusion coeffi 1 cient, and La is the macroscopic absorption cross section. After dividing by 0 and replacing La / O} by K � , we obtain 1 0 .1 5) K % = 0 v
24>
-
In cylindrical coordinates, this becomes
d 20
-
K
0
0 .1 6)
Since K � is positive, Eq. 0 . 1 6) is a modified Bessel equation with the general solution (1 .17) where 10 and K o are zero-order, modified Bessel functions of the first and second kind, respectively. The second term can be eliminated since K o would require the neutron flux to go to infinity at the axis of the fuel rod. Thus, for the flux in the rod we obtain
0 . 18)
The value A can be determined from the boundary condition that is equal to
Thermal neutron flux at fuel rod surface . Mean thermal flux in interior of rod
0 . 1 9)
T H E RMA L A N A L Y S I S O F PWRS
54
With the previous assumptions, for a cylindrical rod we can show that K R 10 ( K--"'o) o R--"F= o o � --
2
0 .20)
11 (K o R o )
For an infinite slab fuel element, we obtain
--�
0 .21)
coth(K o a)
0 .22)
F = K aa where
x = distance from the centerline a = half thickness of the fuel element
0
The computation of K O requires a value of 1 for the fuel. Because dif fusion coefficients are normally not tabulated for fuel materials, it is nec 0 essary to obtain 1 from 0 1 = 3 L tr = 3 L t O
1
[
4 La L a II - /-1 ) 1 - 5 L t + Lt O - II ) + _
( )
]
1 .23
where
L a = macroscopic absorption cross section of fuel L tr = macroscopic transport cross section of fuel L t = total macroscopic cross section of fuel jI = mean value of cosine of scattering angle=2/ (3A) A
=
atomic mass number of fuel.
The previous assumption (that power distribution across the fuel rod is directly proportional to thermal flux) is not entirely correct because, as burnup proceeds, the fuel and poison distribution are no longer uniform. In view of the usual uncertainties in flux levels and in the physical prop erties of the fuel elements, consideration of these nonuniformities is rarely required. Perhaps more important are the limitations on simple diffusion theory. More exact calculations show that diffusion theory tends to under estimate flux depression in the fuel. Most modern computer calculations of flux depression are based on es cape probabilities. In particular, the method of Amouyal al.52 is widely used. This method is carried out only for the equivalent cell; the neutron current is taken to be zero at the outer surface of this cell. However, dif fusion theory is not assumed to hold within the fuel. Amouyal al. cal culate P F as the probability that neutrons produced uniformly within the
et
et
55
POW E R G E N E RATI O N
fuel ultimately escape from the fuel. The expression obtained for a cylin drical fuel rod of radius R o (cm) is -= 1
1
PF
+
(L a F /2. t F HA [1
+
a (L sF / Lt F ) 0 .24)
where
A = 1 - PFo _ � R O �t F PFo
PFo = probability that a neutron escapes without collision from an infinite cylinder of radius R and macroscopic cross section L t F (given by Fig. 1 .3 1 ) macroscopic absorption cross section of the fuel, cm a L F 2.t F = =
total macroscopic cross section of the fuel, cm - 1 Ls F = macroscopic scattering cross section of the fuel, cm a,� = functions of R 0 L t F plotted in Fig. 1 .32.
It may be shown that 1 / PF is equivalent to F. Further details on this method are available from Lamarsh.53 1 .2.6
Distribution of Power Among Fuel, Moderator, and Structure
In addition to axial and radial power distributions, the thermal designer must also be concerned with the fraction of heat generated within the fuel elements. Fission energy is released in the form of kinetic energy of fission fragments, neutron kinetic energy, and gamma and beta rays. The very short range of fission fragments and beta particles ensures that this heat release will take place within the fuel elements.* The energy released during neutron thermalization is released to the moderator. The energy released on neutron capture is largely released within the fuel, but a portion of it is released within the reactor structure. The large mass of fuel produces a substantial self shielding, thus most gammas are captured within the fuel. Table 1 .III quantitatively describes the distribution. A total of 5 MeV of neutron kinetic energy is absorbed during thermal ization. Hence, �2.5% of the fission energy is definitely absorbed by the moderator. This can be taken advantage of in computing the heat that must be transferred by the fuel. The portion of gamma and neutron energy absorbed by the moderator and core structure depends on the specific design of the reactor. In a typical *Approximately 1 00k of the beta energy is converted to gamma energy as bremsstrahlung and should be treated as gamma energy in the division.
56
T H E RMAL A N A L Y S I S O F PWR S
1%
PWR design, somewhat less than of the total power is absorbed within the thermal shield; most of that energy is released by the outer fuel ele ments. In the central region of the core, the gamma energy is roughly dis tributed among the core constituents proportionally to their mass. In a uniform lattice core, this leads to the conclusion that essentially all gammas in the central core region are captured by the fuel elements. In a pressure
1 .0
0.9
1 .0
0.8
0.8
0.7
\
0.6 0 u.. 0...
0.5
0.4
0.3
0.2
�
0
u.. Cl...
\ \
\
0.6
0.4
0
o
f'
"
0.2
\
\
� '"
"'-
r---.... r---...
0.4
0.6
2
�r------.
3 Ro � t F
Fig. 1 .3 1
I
; :-
r--.... .......
......
0.8
1 .0
Ro � tF
0. 1 o
�
4
----t--
5
6
7
The probability, P F ' tha t a neutron escapes without collision i n an infinite cylinder of radius R[) and macroscopic cross-section 2,/ F [from J. Nucl. Energy, 6,
79 (1957) ].
POW E R G E N E RATI O N
0.2 0-
a
0. 1 5 .-�. � �I-
�:t 0. 1
- .
-
co.
- -
- - f- -
f-
0-- -
0.05 �
o 17
V
V
/ V
V
j
�I'
1 .3 2
Parameters
[/"'"
,,7
t- f----I---
- {3
, - 1----- f--
1- -
I /
0:
�
f---
L.---10-
o
Fig.
/
/
....V .
/�
V
'/
l/ V ./
2
RO � t F
5
4
J
and [3 as a function of Ro Ll F [from J. Nucl. fllergy, 6, 79 (1 957) ] .
tube reactor, which has a larger mass of moderator and internal structure, appropriate adjustment should be made. The energy released per fission somewhat depends on fissile material as well as on control material and reactor structure. Walker54 measured the total energy release for various fissile materials in pressure tube reactors of the CANDU design. His results (Table l .1V) indicate a value for 235U fission that is �2% higher than that listed in Table 1 .111.
TAB L E 1 . 1 1 1 Energy B reakdown (MeV/F ission) i n a Water-Cooled Reactor
Time Receiver Fuel element Dispersed among fuel, moderator, and structure TOTAL
Prompt
Fission fragment KE Neutron KE
'Y
1 68 5 7.5 1 80.5
[3
Neutron capture 'Y
Total
Delayed 7
4
6 17
1 75 22.5 197.5
58
T H E RMAL A N A L Y S I S O F PWRS
TAB L E l . 1 V Total Energy* Released per F ission i n Reactors of CA N D U Design 212Th 231U 2YiU 23HU 239p U 241 p U
1 96.2 ± 1 . 1 1 99.7 ± 1 .1 201 . 7 ± 0.6 208.5 ± 1 .1 2 10.7 ± 1 .2 2 13.8 ± 1 .0
MeV MeV MeV MeV MeV MeV
*I ncludes decay of absorption products but not energy of neutrinos.
1 .3 1 .3.1
M ETH ODS FOR EST I MAT I O N OF STEADY-STATE CORE POW E R IN CON V E NT I O N A L PWRs Pi n-by-Pin Diffusion Com putations
To analyze core behavior properly, an accurate prediction of the three dimensional pin-by-pin distribution is needed. While neutron transport cal culations provide the most accurate prediction, no practical methods are available for performing routine three-dimensional pin-by-pin transport computations. Most often, neutron transport methods are used to analyze the behavior of a small region of the core (e.g., a single fuel assembly) in two dimensions. The results of such computations are used as benchmarks from which correction factors for less precise methods can be determined. PWR analyses rely on diffusion theory approximations. However, core wide pin-by-pin analyses in three dimensions are still prohibitively expen sive even when diffusion theory is used. The usual approach uses a synthesis of a two-dimensional pin-by-pin diffusion calculation in the x-y (horizontal) plane and a less detailed one-dimensional axial calculation. This approach assumes separability of the radial and axial flux shapes and hence that three-dimensional peaking can be predicted by a synthesis of two-dimensional models. PWR diffusion calculations usually only consider two-neutron energy groups: fast and thermal. The first step in such calculations is the deter mination of the diffusion constants and cross sections for those energy groups by appropriately collapsing the flux weighted properties based on the neutron spectrum present. The group constants are calculated by such pin-cell codes as GAM55 and THERMOS.56 The second stage of the computations is the use of a two-dimensional diffusion code, such as PDQ,57 to solve the two-group diffusion equation. For the fast neutron groups we have at an interior node
POWE R G E N E RAT I O N
- V 0 1 ( X,y) V
1
= - [VL!J ( x, Y)
+
+
59
LI ( X, Y)
VLh ( x, Y)
0 .25)
For the thermal neutron group we have - V O2 ( X,y)V
where
keff =
0 1 , 02 =
L V L2 =
LsI =
Lf 1 , Lf 2 =
v =
0 .26)
effective multiplication factor of reactor diffusion coefficients for fast and thermal neutrons, respectively macroscopic neutron absorption cross sections for fast and thermal groups, respectively macroscopic cross section for scattering of fast neutrons into thermal energy group macroscopic fast and thermal fission cross sections, respectively average number of neutrons emitted per fission.
Since neutron diffusion is considered only in the x and y directions by Eqs. 0 .25) and ( 1 .26), a correction for leakage in the z direction is made by addition of a term depending on the z direction buckling. Equations 0 .25) and 0 .26) are rewritten in finite difference form for each mesh point considered. The resulting set of linear equations is then usually solved by an inner and outer loop iteration scheme.58,59 The equations are first solved iteratively for the flux distribution by initially assuming a value for keff and solving for the fluxes on the right-hand sides of Eqs. 0 .25) and (1 .26). An improved value for keff can then be obtained using the definition of keff as the ratio of fission neutrons generated in two successive genera tions or, in this case, iterations. The revised estimate of keff is then:
f s y)
( x,y) d x dy
k�?/ ) = ------- I
where
fsy
-
( x,y) d x dy / ( k �ff )
Sf ( x,y) = VL!J ( X,Y)
(n 1) (n)
0 .27)
+ 1) (n)
and + and refer to the (11 and iterations. The improved value of keff and initial set of fluxes is then used to generate a new flux distri-
T H E RMAL A N A L Y S I S OF PWR S
60
keff
con bution. The process continues iteratively until successive values of verge within some preset limit. The most important deficiency of pin-by-pin diffusion methods is that these methods fail to predict the flux correctly in the immediate vicinity of water holes or strong absorbers. To correct for this, pin-cell cross sections are usually adjusted by multiplicative terms called The factors that are used for water holes, burnable absorber rods, and pins immediately surrounding these locations adjust the diffusion theory results so that they provide a reasonable approximation of transport theory results for small core regions. Most pin-by-pin diffusion models do not incorporate any thermal hydraulic feedback since only a two-dimensional representation is used and, in contrast to BWRs, vapor volume fractions during normal operation are small. However, there are some modified versions of PDQ60 that allow for thermal-hydraulic effects through tabular input, which supplies ap proximate corrections that depend on local power levels. It is possible to link thermal calculations directly to the detailed reactor physics of a pin by-pin code, as in the THUNDER code,6 1 but this procedure has been con sidered too time-consuming for general use. This assessment may change as computer capabilities increase. If the variation of power distribution over core life is desired, it is nec essary to couple the diffusion calculations to burnup calculations that de termine the changes in fuel composition. The power distribution and burnup calculations are then repeated in a stepwise manner over the length of the operating cycle. Due allowance must be made for changes in control rod position and in the concentration of dissolved boron.
g factors.
1 . 3 .2 1 .3 . 2 . 1
Nodal Methods Simple Nodal Methods
The necessity for incorporating thermal-hydraulic feedback into BWR neu tronic calculations leads to the use of three-dimensional calculations for this reactor type. Since pin-by-pin calculations in three dimensions are, as we have seen, prohibitively time consuming, it is customary to obtain the over all core flux distribution based on a coarser subdivision of the core. In the radial direction, the BWR core is usually considered on an assembly-by assembly basis. Axial segments on the order of to 1 ft are common. Each three-dimensional segment is considered a node. When such a coarse division of the core is used, the basic assumptions used for the diffusion of thermal neutrons are questionable since the dif fusion length for a thermal neutron is considerably less than the distance between assembly centers. Further, the finite difference approximation of flux gradients is no longer valid. Accordingly, the basic equation for deter mination of the neutron flux distribution is based on the probability that a
0.5
61
POW E R G E N E RAT I O N
neutron born in one location or node will be absorbed in another location or node. In FLARE, the oldest nodal code,62 the neutron conservation equation is written for each node as 0 .28) where
Wm i =
kClJ i = ai = =
probability that neutron born at node m is absorbed at node (transit probability) infinite lattice multiplication factor at spatial node reflection parameter at node eigenvalue thermal neutron flux at node
i
i
i
i.
If three dimensions are being considered, the summation is taken over the six nearest neighbors. In a two-dimensional problem, the summation is taken over the assemblies at the four sides of node The reflection coeffi cient ai has a nonzero value only when the assembly is at the core edge, and one or more adjacent nodes must then be omitted from the summation. In revised versions of FLARE,63 the transit probability, Wm i , which is composed of transport and diffusion parameters, is given by
i.
0 .29) where
M� = migration area = node interval = mixing factor (input parameter) .
Lllm i g' g'
g'
Accurate results from a nodal code require that input parameters (dif ferent can be used for horizontal and vertical directions) and a be fitted so initial agreement is obtained between the nodal code and few-group calculations. If an appropriate regression formula for the effect of voids, burnup, etc., on kx is obtained, the nodal code can then be used to evaluate the effect of thermal conditions on flux and power. Somewhat better agree ment can be obtained by considering coupling with diagonally adjacent nodes as is done in the TRILUX code.64 Although both FLARE and TRILUX were originally devised for BWRs where void effects are of major importance, simple nodal codes were also adapted for PWR use where rapid power estimations were needed.
62
T H E RMAL A N A L Y S I S O F PWR S
g'
Chitkara and Weisman65 used a similar approach for the rapid power es timation needed in fuel shuffling studies. These authors considered to be unity in Eq. 0 .29). They also included the effect of diagonal assemblies by extending the summation of Eq. 0 .28) to include these assemblies. In eval uating Will i for diagonally placed assemblies, Eq. 0 .29) was multiplied by an obliqueness parameter to which they assigned a value of Chitkara and Weisman developed simplified relationships for pi the thermal power
0.7
pi =
0
0 .3 )
where
th (T
Ml ,a
�1 =
thermal flux at given node ratio of fast to thermal flux, assumed constant for purposes of power estimation thermal fission cross sections of 235U and 23t)pU, respectively fast fission cross sections of 235U, 239pU, and 23RU, respectively number densities of 241 pU, 235U, 23RU, and 239pU, respectively fission energies liberated by 241 pU, 235U, 23RU, and 239 pU, respectively.
E E
E 1.04Es' N,
For simplicity, we assume that = and = The number densi 9 R 5 ties of the various fissile isotopes are described in terms of the number density of 23RU and b, the degree of burnup. The degree of burnup is de fined by ( 1 .31 ) where t equals time and a�� equals the thermal absorption cross section of 23SU. At the end of a cycle, btlJl the final degree of burnup is T bt b = b i n + a l;�
fo
where T = cycle length and bi n = degree of burnup at BOC. Chitkara and Weisman6s concluded that bt b could be approximated by 0 .33)
63
POWE R G E N E RATI O N
where
(++ )
(f)
= effective average thermal flux =
= thermal flux at BOC
<Pavg
1
<Pa vg
= average thermal flux across entire core, or radial slice if only a two-dimensional calculation.
- b)
The number density of 235U is then obtained from N
/ Ns = ( N / Ng )O exp( 5 5 where ( N5 / Ns )O is the initial concentration of 235 U. The fissile plutonium concentration is given by
0 .34)
0 .35) where CR is the conversion parameter, which was found to be 0.43 for the San Onofre core. To obtain accurate flux distributions with a nodal code, the user must carefully fit mixing parameters and reflection coefficients to achieve initial agreement with a multigroup calculation for some base loading of a core with the same configuration and power level. Some investigators63 have used formal optimization procedures for this parameter determination. The very rapid simple nodal codes are particularly well adapted to fuel management studies where a large number of fuel loading arrangements need to be examined. An interesting variant of this approach is the so-called "backward diffusion" approach of Chao and HU.66 Since the assembly power distribution is primarily determined by the reactivity, Chao and Hu begin with a desired power distribution. They then solve the diffusion equation backwardly to determine the corresponding core reactivity distri bution. A loading pattern search subsequently matches the available assem blies to the predicted reactivity distribution. When using a simple nodal code for global power distribution, the pin by-pin power in the hot assembly may be obtained by superimposing the hot assembly flux distribution on the global power. This can be done by writing the finite difference approximation for the thermal or two-group neutron diffusion equations for each of the pins in the hot assembly. The boundary conditions can be either a zero flux gradient at the assembly edge or estimated neutron fluxes along the assembly edges obtained by inter polation between the values at the centers of neighboring nodes. 1 .3 . 2 .2
Advanced Nodal Diffusion Computations
Although simple nodal methods were originally used in PWR analysis only to obtain approximate power distributions rapidly, advanced nodal meth-
T H E RMAL ANALYS I S OF PWRS
64
ods that are considerably more accurate are now available. These advanced methods, which directly and rapidly supply a three-dimensional flux dis tribution without the necessity of a radial and axial synthesis, have largely displaced the pin-by-pin diffusion model as the basic neutronic computa tion procedure for PWR design. In common with the simple nodal coupling methods, the advanced nodal diffusion methods factor the core-wide pin power distribution into ( 1 ) a global (homogeneous) power distribution and (2) a local (heterogeneous) power distribution. The relative local power distribution is determined from multigroup transport theory calculations for a single assembly. These calculations, which generally assume a zero-leakage boundary condition, determine a set of pin factors that are subsequently superimposed on the global flux distribution. The transport calculations also provide the homo geneous assembly cross sections required as input by the global power calculations. CASM067,68 and PHOENIX69 are two of the well-known transport-based computer programs that provide appropriate group con stants and pin factors. Since local power peaking around water slots or burnable absorbers is obtained from assembly transport calculations, the previously noted diffusion theory errors are not encountered. To obtain the global power distribution, the core is divided into a series of nodes each of which is considered to be homogeneous. In the usual PWR practice, each assembly is broken into four subsections in the x-y plane and into 6- to 12-in. segments in the (axial) dimension. Diffusion equations are written for both the thermal and fast group neutrons. However, unlike the simple nodal methods where only the average nodal flux is determined, the advanced nodal methods determine the parameters enabling both the average flux and distribution of the flux within the node to be calculated. This is done by determining the parameters of the equations representing the flux distributions in the individual nodes. When the static diffusion equation for every group G is integrated over node K of volume d X d Y d Z, we obtain7o
z
l
dX
]+ l ( K dY
K JCY+ - JCY-
)+ l ( K dZ
K Jcz + - J ez
)
(1 .36)
x)
J �x +
L �,K
K
Jcx + - Jcx -
+ L �,K 4>� = Q�
where
4> � Q�
[K
=
the current on the right (plus face of the node, etc. the average flux in the node the average neutron source i n the node macroscopic absorption cross section for node.
The source term includes fission and scattering in, and therefore depends on the node flux. To establish additional relationships between the nodal
65
POW E R G E N E RATI O N
flux and currents on the nodal faces, the nodal diffusion equation is inte grated over two dimensions at a time. This results in three one-dimensional equations, one for each coordinate. Thus, integrating over �Y� Z results in the following equation dependent on the X coordinate: d K (X ) d x l cx -
+
"" R,K
L.J
c
_ -
K
(1 .37)
II
where the transverse averaged one-dimensional flux is given by Y/2 tl Z / 2 1 dZ d Y � x (X ) = UA YUA Z - tlZ / 2 - AY / 2
I
I
The transverse leakages, L �z (X ) and L � y (X ) , are then 1 1l Z / 2 a d Z - D�- � (X, y,Z ) ]�: �Yfl/ 2 L �y(X ) = A y UZ a - tl Z / 2 y 1 tl / a dY - D�-:- � (X,y, Z ) ]� : ��q/ 2 L � z (X ) = A Uy Z d - tlY / 2
f f
0 .38)
0 .39) 0 .40)
Integration of the one-dimensional flux and transverse leakages over the dependent dimension, and dividing by the dependent dimension, converts the transverse quantities to node-averaged values. Since the foregoing equations involve the transverse leakages, a solution cannot be obtained without an analytical function providing the flux as a function of position. In the so-called "analytic nodal" method,?l each of the one-dimensional equations are solved analytically. In the "nodal expan sion" method, typified by computer programs such as ANC,72 the one dimensional solutions are represented by a polynomial expansion. The single-node solutions must be connected through interface boundary conditions. The boundary conditions imposed can either require interface neutron flux continuity or interface neutron current continuity. The result ing equations are generally termed nodal coupling equations. The simultaneous solution of the nodal equations provides the overall flux and power distribution in the reactor core. To recover the pin powers, the overall distribution equations for the assembly of interest are superimposed on the pin factors determined by the transport calculations for a single as sembly. The resulting powers are found to have an accuracy that is compa rable or superior to the results from pin-by-pin diffusion models?3 The major advantage lies in the fact that they are 30 to 1 00 times more computationally efficient?3 and hence can easily provide three-dimensional results. This ability is particularly important for the newer fuel assembly designs containing axial blankets and/or part length burnable absorber rods.
66
T H E RMAL A N A L Y S I S OF PWRS
1 .3.3
Effective Fast G roup Methods
l.5-group coarsc
mesh approach,
In its simple form, the effective fast group approach, or is an alternative to the simple nodal methods. The 1 .5-group method has the advantage that it does not require empirical mixing parameters. Although much of the initial impetus came from BWR design74 the approach applies equally well to the PWR?5 In the simple effective fast group approach the fast group diffusion equa tion is solved using finite difference equations. The thermal flux is then calculated assuming zero buckling in the thermal group. We assume that thermal group difference equations are not applicable since the 16- to 24cm nodes are several thermal mean-free-paths apart. Therefore, thermal flux is calculated as if the mesh size were infinite for each node, and the thermal leakage from each node is taken as zero. In a two-dimensional calculation where only four adjacent cells are considered, fast flux
0 .41) where So = neutron source in cell 0 Va = volume associated with cell 0
lk
Ak
=
Lg
=
k
distance between mesh point and mesh point 0 area of the boundary separating mesh point and mesh point 0 fast flux at mesh point k fast group removal cross section at
k
O.
Here, Dk is an effective diffusion coefficient defined by - Da Dk ( o o + �\) Dk = DOok + DkoO
-----
0 .42)
i
i.
where D i = diffusion coefficient for fast group at and 0i = width of cell Fast flux is obtained from Eq. 0 .41 ) by using standard inner and outer iteration methods with fission source updates after each iteration. Fission source So is calculated by 0 .43)
O.
where VLro, VLth O = fast and thermal fission source cross sections, respec tively, and
67
POW E R G E N E RATI O N
(1 .44) Lt hO
O.
l - Ai (j) = A i � � + 4
± ��
j
=
1
]
(1 .45)
j,
where �i= flux at the interface between nodes i and and A i = relative weight f�ctor on the point flux of node i. Borreson74 found A i to be in the range of to In the l .5-group approach, the outside boundary condition may be treated by including reflector nodes at the core boundary. The flux is set equal to zero at an appropriate distance into the reflector. The determina tion of reflection coefficients can thus be avoided. Stout and Robinson75 compared l .5-group calculations of the assembly average power distribution to that obtained from a multigroup fine-mesh calculation. They found a maximum error slightly in excess of In this calculation, all number densities were known. They also found that, by combining the 1 .5-group approach with mathematical programming tech niques, optimal fuel management policies were readily determined. More recently, the effective fast group approach has been combined with some of the concepts from the advanced nodal methods. In the NOVA program,76 the source term in Eq. (1 .41) is rewritten in terms of which differs from the normal definition of the multiplication factor, in that it includes the thermal leakage effect to the first order. After the fast neutron flux has been obtained, the thermal flux obtained from Eq. (1 .44) is cor rected on a node-by-node basis for higher order effects. An analytical func tion is then derived from the three-dimensional flux distribution in each node by imposing the requirements that (1) the volume average of the dis tribution function equal the average nodal flux from the diffusion equation and (2) fluxes at the nodal interfaces must be continuous. This approach is considerably simpler than the usual advanced nodal code and is substan tially faster. Chao and Penkrof7 have noted that much more rapidly running ver sions of the effective fast group approach can be obtained by making use of what they call diffusive homogeneity . For PWRs, the variation of the fast neutron diffusion coefficient is small and it may be assumed to be homo geneous throughout the core. They, therefore, rewrite the basic diffusion equation for a given node as
0.7 0.76.
6%.
koo,k*,
T H E RMAL A N A L Y S I S OF PWR S
68
0.46) where
15 =
�a �j
=
reference diffusion coefficient for fast flux reference absorption cross section effective fission operator.
The effective fission operator provides the source term and absorbs the differences due to the deviation of D and La from 15 and �a. Since 15 and fa are considered invariant, the diffusion equation can be written as
0.47)
[G]
where is a square matrix relating each node to the other nodes. This matrix depends only on 15 and the core geometry and the boundary con ditions. Hence, the matrix can be precalculated for a given core ge ometry and stored . That being the case, the conventional inner and outer scheme required for the solution of the diffusion equation is replaced by a single iteration loop. Computer programs based on this approach are suf ficiently fast running so that they can be used for on-line core monitoring. 1 .4 1 .4.1
[G]
POW E R G E N E RAT I ON A N D D I STRI B UTION I N PRESS U RE TU B E-TYPE CORES Overall Power Distribution
In contrast to light-water-moderated reactors, the fuel in heavy water or graphite-moderated reactors is generally not homogeneously distributed throughout the moderator. As described in Sec. 1 . 1 . 1 , the fuel elements in most designs are grouped together in clusters within pressure-containing tubes dispersed throughout the moderator. Nevertheless, an approximation of the overall distribution of thermal neutron flux, which is adequate for thermal design purposes, can be obtained by considering the moderator and fuel as homogeneous. However, we cannot normally consider these reactors to be bare since they are usually surrounded by effective radial reflectors. The effects of such a reflector can be approximately evaluated by considering two neutron energy groups-thermal and fast. For a cylinder of infinite length, thermal flux within the core of a uni formly enriched reactor is given by78
+
Clo (-yr) where A and C are constants, and � ' and -y are properties of the reactor core materials (see Ref. for further discussion). For thin reflectors, the first term is dominant and flux distribution differs only slightly from that
50
69
POWE R G E N E RATI O N
1.33.
of an unreflected core. The change in flux pattern produced by a reflector around an infinite cylinder is illustrated in Fig. A somewhat cruder approximation of the behavior of a reflected core can be obtained by considering all neutrons to be thermal. When this is done, the resulting flux distribution equations have the same form as those for a bare core, but the overall dimensions are replaced by effective dimen sions. For example, the flux of an unperturbed cylindrical core is approxi mated by
=
Le
(1.48) (1.48)
where Re = effective radius of core allowing for radial reflector and Le = effective length of core allowing for axial reflectors. Equation does not predict the slight rise in core flux that occurs adjacent to the re a reasonable estimate of flux flector. However, it can provide (Fig. throughout most of the core. In the common cylindrical core, the large amount of structural material at the ends limits the effectiveness of any axial reflector. A cosine distri bution with Le L is thus often a very good approximation of the axial power variation in a uniformly loaded, unperturbed core. The large diffusion length of neutrons in heavy water permits an accu rate determination of core power and overall flux shape using coarse-mesh
1.3 1)
=
Core Rad i us, r II Effect i vell Radi u s of Refl ected Core , Re Reflector Rod ius
�r I )( �
u:
e 'S
c:
Q)
Z
�
Bare Core F l ux Fig. 1 .3 3
Thermal neutron flux in bare and reflected cylindrical reactors.
70
T H E RMAL A N A L YS I S OF PWR S
finite difference diffusion computations?9 The core is modeled by dividing it into a series of nodes by means of a grid of mesh planes. Cross sections for each mesh volume are obtained from lattice cell cross sections of the fuel bundle and moderator contained within the cell at an appropriate ir radiation. Incremental cross sections are added to represent any other ma terial that may be present in the mesh volume. Sufficiently accurate results are usually obtained with two-neutron energy group calculations using mesh spacings that are typically one lattice pitch by one lattice pitch by one bundle length. The large spacing allows three-dimensional calculations to be readily obtained. Some computer codes allow variable lattice spacing so that smaller mesh sizes can be used where desired. The overall core reactivity and spatial flux distribution in many CANDU reactors are regulated by zone control units. These units are composed of vertical tubes containing light water that run through the calandria per pendicular to the fuel channels. Core reactivity and local flux are controlled by varying the level of light water in the individual tubes. The control rods for CANDU reactors are usually divided into two groups. The often a cadmium annulus between two stainless steel tubes, are usually fully withdrawn from the core. They are inserted only for reactor shutdown. The provide additional power shap ing and compensate for relatively short-term reactivity changes such as those due to xenon transients. In some designs, the adjuster rods are com posed of a number of bundles containing Zircaloy tubes filled with cobalt.
shutoff rods,
1 .4.2
adjuster rods
Power Generation Within Fuel Clusters
The distance between rods within the pressure tube is generally orders of magnitude smaller than that between pressure tubes, resulting in negligible moderation within the pressure tube in comparison to that obtained be tween the tubes. As a first approximation, we therefore can ignore the area within the pressure tube as a source of thermal neutrons and approximate the thermal diffusion equation for thermal neutrons within the pressure tube by a modification of Eq. Here we have
(1. 1 4).
Dav V 2 Laav4> -
=
(1.49)
0
where Dav and Laa V' respectively, represent the diffusion coefficient and macroscopic absorption coefficient based on homogenizing material within the pressure tube. From the development of Sec. we see that this leads to an Io(R) flux distribution across the fuel cluster. Figure illustrates this behavior in fuel clusters of the heavy-water-moderated and -cooled CVTR.8o Since the diffusion coefficient, Dav' equals the reciprocal of 3Ltr, the value of Dav for use in Eq. is best obtained via the sum of the macroscopic transport cross sections for the elements within the fuel cluster.
1.2.5,
(1.49)
1.34
POW E R G E N ERAT I O N
1 . 60 ..-------, (Di stri but ion Across Corners) 1 .50
l AO
Norma l i zed Curve
1 .30
1 . 20 ........---
3 .66 em
o u o
:c >
1 .10
�
- - - -,
r-
- -- - - - ..... ..,
- - - - - - -,
1 .00
0 .90
....L. .,.j 0.80 L-_........L. .0 o
---I.�..&..._ ._ --'-__ 4.0 3.0 2.0 Rad i u s, R (em)
___
Fig.
1 .34
""O'____'
Flux distribution across radial axis of CVTR assembly (from Ref. 80).
The Io (R) flux distribution within the fuel cluster is clearly only a rough approximation. Highly accurate values of the flux and power within the cluster may be obtained with fine-mesh computations based on neutron transport theory. Less accurate, but acceptable, flux and power distributions can be obtained by fine-mesh multigroup diffusion calculations. is Flux depression within the fuel assembly, often called the Superimposed on the normal radial flux variation across the reactor as a who le. We can then write
fine flux dip,
T H E RMAL A N A L Y S I S O F PWR S
72
Ff{ = F� X F�
(1 .50)
where
FNRn
Mean flux in the hot fuel rod Mean flux in the hot fuel cluster Mean flux in the hot fuel cluster Mean flux in the average fuel cluster
In designing the CVTR reactor, F{j was assigned a value80 of 1 .25 (Fig. 1 .34). Thermal flux within a particular fuel rod will vary with the position of the rods in the cluster and, as we proceed around a fuel rod, the value of R or the radial distance from the pressure tube center varies. In addition, for any tube not in the center of the core, there is an overall flux pattern effect on flux distribution within the cluster. This can be significant in a small core such as that used in the CVTR.sO Thus, for any rod, we can show that = f (r,e)
( 1 .51 )
To simplify temperature distribution calculation, distribution for a given rod is often approximated by a function of the form = C1 + C2r
cose
(1 .52)
The approximate function is made to fit actual distribution at the center of the rod and at the point on the circumference with the peak heat flux.sO In a large core (CANDU designs), variation of the overall flux across a single pressure tube is generally small and angular power variation is of lesser importance. 1 .4.3
Effect of Refuel ing Procedu res
As previously noted, the CANDU reactor is the major PWR pressure tube reactor type of commercial interest. In this design, the large cylindrical cal andria is usually placed with its axis horizontal and is pierced by a large number of once-through horizontal pressure tubes. Since these reactors are fueled with natural uranium, maximum burnups are low (below 10,000 MWd / l 000 kgU) and frequent refueling is necessary. Excessive downtime can be avoided by using on-power refueling. Short fuel assemblies are used so that each channel contains a number of assemblies. When it is desired to add fuel to a channel, specially designed refueling machines are connected to each end of the channel. Fresh fuel assemblies are added at one end of the channel, while the refueling machine removes burned assemblies at the other end. By always adding fuel at the same end of a given channel, the fuel assemblies gradually move across the
POWE R G E N E RATI O N
reactor. Thus, there is a significant burnup gradient along the axis of a given channel and, hence, axial power generation will not follow a simple cosine distribution. To prevent a power tilt in one direction, half the pressure tubes are loaded from one end of the reactor, while the others are loaded from the opposite end. Adjacent tubes are always loaded in opposite directions. At BOL, a fuel assembly will be at the edge of the core and generate a low power. Later in life, it will see the high flux in the center of the reactor and operate at its maximum power. At EOL, it will again be near the edge of the core and at low power. A typical fuel assembly operating history is shown in Fig. 1 .35. A realistic estimate of axial power distribution requires that variation in fuel burnup be considered. Since the burnup distribution is not axially con tinuous, axial power in a given channel will show a stepwise variation. The objectives of the usual CANDU refueling procedures8 1 are to achieve an optimal radial power distribution and to minimize refueling machine relocations. To achieve these results using natural uranium as the fuel, the core is divided into radial zones (usually two), each of which is character ized by an exit fuel burnup and a bundle shift scheme. The bundle shift scheme is designed to insert the maximum number of fresh fuel assemblies into a given channel during refueling without causing excessive power peaking or reactor control problems. In the central region of a large core with channels containing 12 bundles, the typical exit burnup would be set slightly in excess of MWd/ tonne and bundles would be added at a time. The size of the inner region is set so as to keep the maximum flux to-average flux ratio within a prescribed limit. In the outer region, exit burnup would be of the order of MWd / tonne and fresh bundles would be added at a time. If a CANDU reactor is to be fueled using fuel slightly enriched with 235U or Pu, a revised refueling procedure is required to prevent axial flux dis tortions and reductions in the reactivity worth of control and safety rods. The method proposed for slightly enriched fuel81 is the so-called "check-
7000
8
6000
4
r-
...
Q)
�
o
a..
o
Fig.
1 .35
I
Bu rnup
_
Typical operating history of CANDU fuel assembly [from f. NlIcI. Appl. Tech . , 9, 1 45 (1 970) 1 .
T H E RMAL A N A LY S I S OF PWRS
erboard" fuel management scheme (CFS) . In CFS, the core is divided into three radial zones. In the central zone, a checkerboard arrangement of bun dle shifts is used . For example, in a group of four adjacent channels, two would undergo a two-bundle shift and two would undergo a six-bundle shift. The two outer core zones would have a constant bundle shift through out each zone. 1 .5
TRA N S I E NT POW E R G E N E RAT I O N A N D D I STRI B UT I O N
Reactor power distributions during transients are computed via an elabo ration of steady-state diffusion theory. In the most rigorous approach, the neutron diffusion equations [Eqs. and for each control volume are placed in the transient form by the addition of terms for neutron ac cumulation and emission of delayed neutrons. These equations must be coupled to computations of xenon, samarium, and delayed neutron pre cursor concentration changes. A short time step is assumed and the equa tions are solved for the neutron flux, nuclide concentrations, and power distribution at the end of the time interval. Alternatively, simplified ap proaches, such as those based on the point kinetics equation, can be used . These approaches are discussed in greater detail in Sec. where accident analysis techniques are described. In the present section, the discussion is confined to the problems introduced by load changes and heat generation after shutdown.
0 .25) 0.26)]
6.2.2
1 . 5 .1
Power D istribution Fol lowing a Load Change
As previously noted, in a chemical-shim-controlled core, the control rods are nearly fully withdrawn at full power. However, if the plant is to be load following, rods would normally be inserted during part-load opera tion. Improper positioning of these rods for part-load operation can lead to an unacceptably high axial power peaking on return to full power. When control rods are partially inserted, there is a spatial change in fission product distribution. The distribution of 135Xe is of particular im portance because of its high yield and cross section (3 X 106 b). Randall and St. John82 recognized this problem and pointed out that changes in xenon distribution following load changes can result in significant power shifts. More recent studies confirm this viewpoint.50 If all the reactivity change required for low-power operation is compen sated for by rod motion, a deep insertion of the rods is required. This causes the flux to peak in the bottom half of the core and leads to reduced con centrations of 1 35Xe and its precursor, 1351, in the top half of the core. On return to full power by rod withdrawal, the initial 1 35Xe concentration would largely be determined by the concentration of 1351 set at low-power
POW E R G E N E RAT I O N
1.36,
operation. The power now peaks in the upper portion of the core because of the lower 135Xe concentration there. This behavior, illustrated in Fig. results in a higher axial peaking factor than that obtained under steady state conditions. The concept of has been used to avoid in creased axial peaking due to xenon transients.49 The axial offset (AO) can be defined as
constant axial offset control AO =
(1.53)
P� - P � P � + P 'a
where P � = total power generated in the upper half of the core and P'a = total power generated in the bottom half of the core. When a core operates at full power with all rods withdrawn, there is generally an axial offset of somewhere between and (between and on all but the first cycle). Axial offset varies slowly with lifetime. The equilibrium value with rods withdrawn at any given point in life is called the "target" axial offset. It has been found that if AO were required to remain at the target value through all load changes, then on return to full power the axial peaking factor is not significantly changed. If all the changes in reactivity occasioned by a load reduction were com pensated for by boron addition, there would be an upward shift in core
0.0
0.05
0.1
0.0
Tra n si e nt Powe r D i stri bu tion I mmedi ate l y After Rod W i t hdrawa l
t
/-�
-_...._ . -
\
\
L
Fig.
1 .36
Axial power shifts after control rod motion.
D i stri buti on
76
T H E RMAL A N A LYS I S OF PWR S
power; i.e., AO would become positive. This shift arises because of the local negative moderator coefficient effect and the larger temperature change near the top of the core compared to that in the bottom portion. Control rods are inserted at a level sufficient to prevent this power shift, and any additional reactivity reduction is achieved by the addition of boric acid. By using a combination of rod motion and boric acid concentration changes, load changes on the order of O.3 % / min can be accommodated. This rate of power change is sufficient to handle the daily load cycle to which a plant may need to respond. Axial offset can be readily monitored by the reactor operator. In one scheme, ionization chambers that run the full length of the core are placed at four locations around the core circumference. Each chamber is split into an upper and lower portion and separate signals obtained from each sec tion. Electronic comparison of these signals allows the axial offset to be read directly by the operator. The concept of constant axial offset, which minimizes changes in power shape during core lifetime, is similar in some respects to the Haling prin ciple. 83 This principle has been widely used for setting the optimum axial flux shape in BWRs. Haling states that rods compensating for reactivity changes with lifetime should be programmed to keep the axial power shape invariant over the in-core life of the fuel. Although constant axial offset control, which will typically restrict axial offsets to between - 12 and + 3%, allows peaking limits to be satisfied, operation outside these limits can be beneficial. Following a reactor trip or shutdown of short duration the rate at which full power can be attained is significantly restricted by tight offset limits. To alleviate this problem, most plants now use In this approach, it is recognized that although Ffj increases with decreasing power, the power decreases faster than Ffj . This fact, together with additional margin during transients disclosed by more sophisticated analytical methods, means that wider AO limits are permissible at reduced power levels. In most cases, a target AO is established but the actual axial offset limits are now specified by a figure similar to Fig. 1 .37 Since much wider axial offset limits are allowed at low power, operational problems are simplified. However, good operational practice calls for the return to the target axial offset as soon as practicable. The AO limits are determined by considering the power distributions resulting from rod configurations producing a given axial offset. Values of Ffj are computed for various power levels. An axial offset is unacceptable at a particular power level if it produces a peak power at any elevation tha t is in excess of the limiting peak power versus elevation curve. As seen in Fig. 1 .34, positive axial offsets (flux skewed to top of core) are more tightly restricted than negative offsets. This is caused by the CHF limit, which is more difficult to meet in the upper portion of the core where coolant en thalpies are higher.
relaxed axial offset contro1.84
(z)
POW E R G E N E RATION
------
1 20
1 00 ,r-..
cfl
�
80
�
60
U
40 -
J..< Q)
0 p...
J..< 0
0 Q) p::
�
20 -
0
-40
__
-30
L-
-20
-10
----1__
0
_�__ ___ -1
10
20
30
Axial O f fset ( % ) Fig. 1 .37
Typical relationship between maximum allowable power and axial offset.
Limitations on the rate at which boric acid can be added make it difficult for axial offset limitations to be maintained using full-length rods if a rapid (e.g., 5%/min) load change must be accommodated. Some plants have been equipped with "part-length" control rods in an effort to achieve the rapid load following required when a plant is part of the system' s spinning re serve. A part-length rod contains poison material only in the lower portion of the control elements and an inert material, such as Al20 , in the upper 3 portion. If these rods are deeply inserted, they can depress the flux in the lower portion of the core and, thus, produce a more acceptable axial offset. Although it has been possible to meet the peak axial requirements for steady-state operation, such operation can lead to high local stresses on fuel cladding. Such stresses can be calculated as being in excess of those stress levels capable of causing cladding failure by stress corrosion cracking in the presence of iodine. Little use is now made of part-length rods and rapid load follow is more often accomplished by means of "gray" control rods.85,86 A standard or "black" control rod cluster in a 17 X 17 rod assembly will typically contain rods of highly absorbing material (i.e., Ag-In-Cd, boron carbide). In the "Mode G" approach,85,86 a gray cluster would typically contain only highly absorbing rods but also 1 6 stainless-steel rods. Since the reactivity worth of such a cluster is substantially reduced, its motion will have a smaller impact on the power distribution. A large (1300-MWth) core would
24
8
78
T H E RMAL A NA LY S I S O F PWRS
28
typically have gray clusters arranged in four banks. Typically, an addi tional 25 highly absorbing rod clusters would also be present. Seventeen of these rod clusters would usually be used for shut down and the other used as a reactivity control bank (bank R) . To assure that the gray rods are available for unexpected load increases from a scheduled power level, their position is determined solely by the power level. These banks do not react to reactivity changes such as those due to xenon poisoning. These changes are compensated for by the motion of bank R. However, the maneuvering range of this bank is restricted to a narrow band at the top of the core. At this elevation small motions of bank R have little effect on the power distribution. The gray banks are inserted sequentially in the core as the power level is reduced. However, the insertion pattern calls for the overlapping of banks so that the next bank is partially inserted before the previous bank is fully inserted. Because the gray banks produce less radial distortion than insertion of a highly absorbing bank, wider axial offset limits may be used and rapid load follow is possible. In some new plants, the ex-core two-region ionization chambers are re placed by units with a number of axial regions. The readings of these units, together with accurate rod position information, allow the power distri bution at a given time to be reconstructed. The allowable operating range then depends on the actual power distribution existing in the core and is wider than that allowed by the offset limits set on the basis of the most adverse conditions. Load-follow operation is possible using all highly absorbing rods pro viding the control rod banks are properly configured. In the so-called "MSHIM" approachs7 a fraction of the usual control rods is regrouped to create a set of three relatively low worth banks (M banks) and one high worth bank (A/O bank). The remainder of the rods are designated as shut down rods. The M banks, which are 75% inserted at full power, are inserted or withdrawn to compensate for the reactivity change with power. If the M bank movement produces an axial offset outside of the allowable range, the A/O bank (normally in the upper 15% of the core) is adjusted to com pensate for this. Additional M bank movement may then be required to counteract the reactivity change due to the A/O bank movement. Both the MSHIM and Mode G load-following approaches require the continuous insertion of a high worth control rod bank. This causes local burnup shadowing and some asymmetry in the steady-state power shape. In addition, withdrawal of this bank to meet a rapid increase in power may distort the power distribution. Park and Christensod�8 have proposed a revised approach that avoids this difficulty. In their "LG" approach a light gray bank and moderator temperature are used to control negative axial offset, while a high-worth bank is used to control positive AO. Insertion of the high worth bank is not required at steady-state full power. This bank
8
79
POWE R G E N E RATI O N
is needed only when there is a pronounced skew of the power toward the upper portion of the core. Load following complicates plant safeguards studies. A reliable accident analysis begins with power distribution in the core at the onset of the ac cident. Since an accident could occur at any time and since the exact power distribution is a function of the power-time history of the core, there are a very large number of power distributions possible at the onset of an acci dent in a load-following plant. It becomes impractical to analyze each pos sible accident for all possible power distributions. One way of avoiding repetitive accident analyses is to construct an envelope of peak local power density (kW 1ft) versus core height.5o An attempt is made to determine peak power levels under all anticipated conditions by analytically simulating all anticipated load-following behavior over core lifetime. From these many power distribution studies, the worst peak power at each axial position is selected. The synthesis method, described in Sec. is often used for this determination. These powers, which form an upper envelope of all possible powers, are then used to describe core condition at the beginning of the accident. When accident analysis shows that the core design is safe for these conditions, then the core is clearly safe for the less severe conditions that would actually exist. This approach is particularly useful in LOeA analyses.
1.2.4,
1 .5 . 2 1 .5 . 2 . 1
Power Generation D u ring Shutdown Fission Power Generation
The rate of power decrease during a reactor shutdown is of concern to the thermal-hydraulic designer, since it affects fuel temperature during accident situations. We consider the situation where the reactor is shut down after operating at steady state for an appreciable period. During steady-state op eration, the effective multiplication factor, keff' must be unity. For startup, it must be greater than unity. For shutdown, it must be less than unity. It is useful to define an excess multiplication factor kex ' where
(1.54)
Thus, kex is positive for a supercritical reactor and negative for a subcritical reactor. If all neutrons can be considered emitted promptly at the time of fission, the time behavior of the bare thermal reactor is given by
0.55)
where
0.55)
kcx
Equation can be considered valid immediately after the change. For longer periods, the effects of delayed neutron groups must be consid-
T H E RMAL A N A L Y S I S OF PWR S
80
ered. If the delayed neutrons are approximated by a single group with a decay constant A. equal to the weighted average for the five groups, for small reactivity changes we obtain
{
)
[
A P 't ( f3 ---,- ' )t --P_ p'
(
] } 0 .56)
where f3 is the total fraction of delayed neutrons and P ' the reactivity, is defined as
0.57)
, kex keff
P =-
8
The equation is valid only when ( f3 - p ' ) is positive. Since I is on the order of magnitude of 10 - 3 s and A is 0 0 s - l , the flux during shutdown is the sum of two positive terms ( p ' negative), one of which decreases much more rapidly than the other. The flux and, hence, power initially decrease rapidly and then fall off slowly in accordance with the first term within the bracket; subsequent slow decay is due to the effect of delayed neutrons. The usual reactor shutdown is brought about by large reductions in reac tivity. For this condition and considering all five groups of delayed neu trons, the neutron flux is closely approximated by �
.
-
exp( - t A l ) + A exp( t A. ) + 2 2 X exp( t A S ) + A exp( t / 1 ) ] 6
-
-
+ As
0.58)
Here, A are the decay constants for the five groups of delayed neutrons. For large reactivity decreases, the rate of flux decay is independent of the reactivity change. Noting the very small value of I and examining the values of A in Table 1 .V (Ref. we see that initially there is a rapid decrease in flux due to the last term, but after a short time, all terms become negligible except those with the smallest value of A. Flux then decays exponentially with a period corresponding to the maximum 1 / A, or �80 s. Based on the foregoing, a good approximation of the fission energy power following a large reactivity decrease is a sudden decrease (prompt jump) followed by a relatively slow exponential decay. The residual power of a light-water-moderated reactor shutdown, following sustained opera-
89),
TAB L E 1 .V Delayed Neutrons in Thermal Fission of 2 35U
Mean Life
Decay Constant [>.. (s - 1 )]
0.62 2.19 6.50 3 1 .7 80.2
1 .61 0.456 0.151 0.03 1 5 0.0124
Fraction «(3c ) 0.00084 0.0024 0.0021 0.00 1 7 0.00026
81
POWE R G E N E RAT I O N
tion at constant power, can be approximated over the time period during which decay heat is important by
P'/Pb=0. 1 5
t
exp( - O. l
(1.59)
t)
where is in seconds. Since the neutron lifetime on which the decay constant depends is largely determined by the reactor moderator, Eg. should only be used for LWRs. For heavy-water-moderated reactors, a decay constant of should replace the Also note that Eg. does not apply to a plutonium fueled core. The delayed neutron fraction for 239pU is only % and, hence, the residual power will be about one-third that of 235U fuel.
0. 1 .
1 .5 . 2.2
(1.59)
(1.59)
0.06 �0.21
Oecay Heat
(1)
(2)
In addition to residual fission energy, there are two other major heat sources: fission product radioactive decay and neutron capture prod uct radioactive decay. Each of these sources must be considered . Perkins and King,':!o Smith,9 1 and Shure,92 among others, considered fis sion product decay heat generation. The design standard93 was based on Shure' s work.92 Decay heat generation can be determined from the curves of Fig. for reactors with an infinitely long operating time. The
1973
1.38
1 0- 1 If>
o
1 0- 4 L--L��� 1 05 1 04
F ig.
1.38
1 06 T i me A fte r S hu tdown, s
1 09
���Ull--��LU���LLWU������
__
Standard fission product decay heat curve for uranium-fueled thermal reactors (infinite operating time assumed).
T H ERMAL A N A L Y S I S OF PWRS
82
ordinate of this figure, P ' / P b, is the ratio of decay power to operating power. An upper design limit is obtained from these curves since they were obtained by adding an additional to Shure' s values to account for uncertainties. Schenter and England94 compared Shure ' s curve to more re cent decay heat computations and experiments. They concluded that the Shure + curve is conservative. Nevertheless, the NRC regulations for evaluation of emergency core cooling systems95 require that the curve of Fig. be used . This requirement remained in force at the time this text was prepared As a safety evaluation requirement, the curve has the advantage of being conservative and easy to use. Schenter and Schmittroth96 point out that Shure ' s decay heat evaluation is based on 235U fission and decay heat depends on the fissionable isotope used. They observe that decay heat from 24 1 pU fission products is very close to that for 235U fission products for short cooling times and slightly below at long cooling times. Fission products of 239 pU produce heats that are below the 235U curve at zero time and the same at s. Decay heat resulting from fast fission of 238U is above the 235U at zero time and slightly below at s. However, 238U fast fission is only a very small fraction of the total fissions. The uncertainty included in the design curve would appear to be very conservative and encompass any effects due to fissioning isotopes other than 235U. If the residual heat generation after several days of operation were of interest, the fact that the reactor has operated for a finite time rather than an infinite time becomes important. Obenshain and Foderaro97 state that residual heat for a finite time of operation (To) can be found by subtracting the value of (t + To ) from the value for infinite operation with a true decay time of t. That is,
20%
1.38
�
20%
(1994).
15%
1000
20%
�20%
�
1000
(1.60)
where
(
P' decay heat ratio for a reactor that has operated for time t t ) = t and been shut down for time t 5 Pb 0' P' ( decay heat ratio for a reactor operated for an infinite t = time and shut down for time t 5 Pb CXJ
( 1993
P' Pb
'
5
)
5
heat ratio for a reactor operated for an infinite CXJ' t0 + t5 ) = decay time and shut down for time (to + t5).
(1.60).
The decay heat standard98 prescribes a more accurate and elaborate procedure in place of Eq. Also note that for decay times in excess of min, the contribution of residual fission energy generally can be ignored. The American standard was revised in and again in to reflect the more recent information. The most important of these revisions was the
1
1979
199399
83
POW E R G E N E RATI O N
inclusion of separate estimates of decay power from thermal fissions of 235U, 239pU, 241 pU, and fast fission of 238U. The values of Fj (t,oo), the decay power per fission (MeV l fission), after an infinite reactor operating time and cooling time t, which are contained in the 1 993 standard, are summarized in Table l .VI. Note that when this table is used, the decay power depends on the isotopic composition of the core fuel and the fraction of the total power generated by each isotope. To obtain the decay power (MeV Is) from Table I .VI, the tabulated values must be multiplied by the reactor fission rate (fission I s) prior to shutdown. The values from the revised standards presented in Table l .VI are based on the assumption that there is no neutron capture in the fission products. To account for this effect, the values from the table are multiplied by a factor, designated G (t), which corrects for this capture. The values of G (t) presented in Table l .VII are the maximum values believed to apply. This is done to assure that conservative decay powers are obtained. In addition to accounting for the decay heat from fission products, the revised American standard requires that the heating due to actinide decay be included. This contribution, for reactors that have been operating for long periods, is given by: F239 u (t) = E 2 u R exp( - X- t) 39 1 X- 1 F2 N p(t) = E 2-19NpR __ _ exp( - X-2 t) - � eXP( X- t) 39 1 X- l - X-2 X- l X-2
[( - )
where
]
(l .61 ) 0 .62)
t = time after shutdown, s E 239u = average energy from the decay of one 239U atom = 0.474 MeV E2 9NP = average from the decay of one 239Np atom = 0.419 MeV 3 R = atoms of 239U produced per second per fission per second evaluated for reactor composition at the time of shutdown X- l = decay constant for 239U ( = 4.9 X 1 0 - 4 S - 1 ) X-2 = decay constant for 239Np ( = 3.41 X 1 0 - 6 S - 1 )
1
Table 1 .VI F ission Produ ct Decay Power*
Isotope 235U 239p U 238U 241 p U
1 0°
1 01
1 02
1 03
1 04
1 05
1 06
1 07
1 2.38 1 0.27 1 4.76 12.10
9.50 8.25 1 0.36 9.13
6.20 5.69 6.38 5.78
3 .80 3.52 3.71 3.44
1 .91 1 .73 1 .80 1 .65
9.73 x 10 1 9.47 x 1 0 9.33 x 1 0 9.105 x 1 0 -
5.55 X 1 0 - 1 5.14x 105.10 x 1 0 - 1 5 .02 x 1 0
2.50 x 1 0 1 2.34 x 1 0 2.23 x 1 0 2.35 x 1 0 -
* In MeV I fission at t seconds after an infinite reactor operating time at thermal fission in the absence of neutron capture in fission products (1 993 ANS standard).
84
T � ERMAL ANALYSIS OF PWRS
TA B L E 1 .VI I Factor Accounting for Effect of N eutron Capt u re by Fission Products on Decay Heat f(5)
G (t)
f(5 )
G (t) 1 .064 1 . 1 24 1 . 1 24 1.181
1 0° 1 01 1 02 1 0)
1 .020 1 .022 1 .023 1 .033
1 04 l Os 1 06 1 07
and F has units of (MeV / s) per (fission/ s). The decay heat power contrib uted by these heavy elements, PdH E ( t ) , is then calculated from p
Pd I IE( t ) = Q [ FZ39u(t)
0 .63)
+ FZJ9 N P ( t ) ]
where P is taken as the maximum reactor power during the operating his tory and Q is the effective energy release per fission evaluated for the re actor composition at the time of shutdown. Besides the American standard on decay heat, a German standard has been adoptedl llO and standards have been proposed by the Japan Atomic Energy Research Institute1 01 and the International Standards Or ganization ( 1 50 ) . 1 02 The latter is for use in countries desiring a standard and not having one of their own. Dickens et al. 1 03 have compared these standards and find a substantial degree of similarity between them, but minor differences at a number of points. An alternative approach to computation of decay heat is the use of one of the computer programs that determines the concentration of the individ ual fission products and their hea t generation as a function of operating and shu tdown times. The ORIGEN code 10 4 is, perhaps, the best known of these programs. Decay heat due to capture in control material varies greatly depending on the nature of the absorber present in the core during power operation. Decay heat due to silver-indium-cadmium control rods would be more sig nificant than that due to boron-stainless-steel rods. In any case, the capture product decay heat is substantially less than that due to capture in 23HU , but no general expression for this heat generation is available. The contribution from capture in the structure is small. It is generally satisfactory to ignore this. By combining the estimates from Fig. with those of Eqs. 0 .59) and we estimate the total power generation at time t after a reactor shut down. Figure shows the relative power generation for a typical light water-moderated reactor, where the heat released to the coolant is appreciably greater than the heat generated, due to the release of heat stored in the fuel. Means for taking this into account are discussed in Sec. 2.6.
(992)
(1.63),
1.39
1.38
85
POW E R G E N E RATI O N
Rods 85% Inse rted E ffect i ve Scram
1 00
x" Q) :::> � 0
80
Average Heat
2
� >
� '';:
- 0
Flux 60
Q) ' -
...... .E ....
Q)
.....
0
� "e
40
0.. 0
20 0 0
2
4
8
6 Ti me,
Fig. 1 .39
1 .6 1 .6.1
10
12
14
s
Relative power level and heat flux after a scram [from Westinghouse Engilleer, 25, 5, 1 45 ( 1 965)] .
H EAT G E N E RATI O N I N REACTOR STR UCT U RE A N D MODERATOR Structu re
Heat is produced in the fuel by absorption of beta rays and the kinetic energy of the fission fragments, while outside the fuel, heat is generated primarily by gamma-ray absorption. Gamma rays come from gammas re leased in fission, those released by decaying fission products, or those formed by absorption of neutrons in the core materials. The variation of the uncollided flux of gamma rays <po of a given energy through a slab with a source on one side is given by where
exp(
-
/-LjX)
0 .64)
Sa; = flux
at the surface, MeV /s cm2 x = distance from surface, em /Li = attenuation coefficient, em
The associated heat production is then
q? = /LlI; Sa ; exp( /-Lj x)
0 .65)
where /-La ; = energy absorption coefficient (em - 1 ) and q? = heat production (MeV / s cm3). In a reactor where we have a source emitting rays of a num-
86
TH E RMAL ANALYSIS OF PWRS
ber of different energies, the total heat production due to uncollided flux is the sum of heat production due to each energy level.
q? =
fJ..l l1 Sa [
exp(
-
fJ.. 1 X) +
fJ.1. l2S112
exp(
-
(1 .66)
fJ..2X) +
The actual heat production is greater than indicated by Eq. 0 .66) due to absorption of secondary gammas from Compton scattering. Additional heat production is computed by buildup factors such that total heat produc tion is given by
Bai q = q� Bal + q� B1l2 q� Ba3
q
+
+
(1 .67)
fJ..x
The buildup factors differ for different materials and are a function of and the energy level of the gamma rays. Rockwelll OS gives a graphic pres entation of these. In a reactor core that has been operating for 1 00 hr or more, the gamma ray energy from both fission and fission products is assumed to be directly proportional to the local power density. Values for gamma-ray energy pro duction from fission and fission products, as well as energy attenuation coefficients, are summarized by McLain and Martens.H16 Where gamma flux leaving the core must be determined, power density near the surface of the core can be used as an approximation, since high self-shielding of the core minimizes contributions from other regions. The variation of neutron flux through a thin slab with a source on one side is of the same form as that for gamma rays where
exp(
-
2:x)
0 .68)
local neutron flux at neutron flux at surface of slab neutron absorption coefficient, cm -
In a structural material consisting of heavy elements, neutron heat produc tion is primarily due to absorption of gammas produced by neutron inter due to capture gamma rays action with the structure. Heat production of a given energy level, is therefore directly proportional to the neutron flux and a function of and total slab thickness. If the neutron flux is given by Eq. 0 .56), then
ql l ,
fJ.a. ' x, q�=KEf(E) fJ.l. (2: / fJ..)
where
�= q f(E)L =
,
0 .69)
/
heat production due to capture gamma-ray absorption, W cm3 fraction of captures producing gamma rays of energy total slab thickness
E
87
POW E R G E N E R AT I O N
K = 0.8 X 10 - 13 Ws/MeV !-L = linear attenuation coefficient for gammas of energy E, cm !-La = energy absorption coefficient for gammas of energy E, cm and F (b,a) = 1
fo exp(at)E1 (t) dt; b
0 .70)
where b and a are the appropriate arguments required by Eq. 0 .69). Func tion Fl can be evaluated from Rockwell' s graphs. 107 Equation 0 .69) neglects buildup of gamma rays due to Compton scattering; therefore, these results should be modified by application of the appropriate buildup factor for thick elements. Total heat generation is then the sum of these values and those obtained from Eq. 0 .67). For steel slab thicknesses of < 1 .5 em, it is usually satisfactory to assume a constant heat production due to neutron absorption. In this range, buildup factors approximately compensate for neutron flux attenuation. Furthermore, in the slabs with thicknesses between 1 .5 and 6 em, absorption of capture gammas account for < 1 0% of total heat production. This heat production can be approximated as a constant fraction of total heat pro duction. Note that in a pressure vessel wall adjacent to a shield tank, the neutron flux begins to rise near the shield tank due to the reflected flux. Here, a better approximation would be constant heat production due to neutron absorption. 1 .6.2
Moderator
While heat generation in the reactor structure is primarily from gamma-ray absorption, heat generation within the moderator is primarily from neutron thermalization through elastic scattering. For elastic scattering qlll(X), the volumetric heat generation rate at some location x can be approximated by 1 08 0 .71 ) where Ls = macroscopic elastic scattering cross section of the fast neutron flux /x) and � = average energy decrement per collision. 1 .7
T H E RMAL DESI G N BAS I S
In view of the many variables that affect power generation within the re actor fuel, it is clear that an accurate computation of the relative power distribution in the reactor is quite complicated. The simple approximations presented in this chapter for uniform, unperturbed cores can be regarded as useful in providing very rough estimates for preliminary design. How-
88
T H E RMAL ANALYSIS O F PWR S
ever, any actual design must be based on realistic power distributions ob tained from the nuclear designer. The major results of the nuclear calculations can be relayed to the thermal designer in terms of nuclear hot channel factors. These factors, with an estimate of the axial power distribution shape, can serve as a basis for initial plant thermal design. Because of the wide use of chemical shim control, simple relationships are often adequate to describe axial power shape. Refined analyses of plant performance may wish to consider thermal and nuclear effects simultaneously. This can be accomplished by linking the approximate methods for power estimation from Sees. 1 .3.2 and 1 .3.3 with the appropriate thermal design procedures.
References 2. 3. 4. 5. 6. 7 8. 9. 10.
12. 13. 14. 15. 16. 17 1 8. 19. 20.
G. H. Farbman, "The Pressurized Water Reactor," Paper No. 65-15, Westinghouse Nu clear Power Seminar (1965). J. K. Pickard, Ed., Nuclear Power Reactors, pp. 70-73, D. Van Nostrand Company, Prince ton, NJ (1957). H. J. von Hollen and C. F. Currey, "Design Features of San Onofre Nuclear Station," Westinghollse Engineer, 25, 5, 145 (1965). E. U. Powell and R. M. Harper, "R. E. Gina Nuclear Power Plant Unit No. 1 ," Proc. A m . Power Conj., 30, 298 (1968). Stuart McLaine and J. H. Martens, Eds., Reactor Handbook, 2nd ed., Vol. IV, pp. 557-560, Wiley-Interscience Publishers, New York (1 964). W. D. Fletcher and D. D. Malinowski, "Operating Experience with Westinghouse Steam Generators," Nucl. Technol., 28, 356 (1976). M. A. Styrikovitch, V S. Polonsky, G. V Tsiklauri, Two-Phase Cooling and Corrosion in Nllclear Power Plants, Hemisphere Publishing Corp., Washington, DC (1987). T Matsuoka, "A Simplified Japanese PWR," Nucl. Eng. Inti. 36, 443, 47 (1991 ). D. K. Davies, "Nuclear Steam Generators," Proc. A m . Power Conj., 27, 310 (1965). D. Kalafati, "Thermodynamic Cycles of Nuclear Power Stations," Israel Program for Sci entific Translation (1965). F. A. Artusa, "Turbines and Cycles for Nuclear Power Plant Application," Proc. Am. Power Conj., 29, 280 (1967). M. M. EI-Wakil, Powerplant Technology, McGraw-Hill Book Co., New York (1984). J. Weisman and L. E. Eckart, Modern Power Plant Engineering, Prentice Hall, Englewood Cliffs, NJ (1985). "Consolidated Nuclear Steam Generator for Merchant Ship Application," BAW-1243, Babcock & Wilcox Company (Aug. 1962). International Atomic Energy Agency, Directory of Nuclear Reactors, Vol. IV, Power Reactors, pp. 39-45, Vienna (1 962). International Atomic Energy Agency, Directory of Nuclear Reactors, Vol. IV, Power Reactors, pp. 1 63-168, Vienna (1962). A. Gonzalez, K. Frischengruber, J. Recalde, R. Solanilla, and R. Vanzulli, "Argos PHWR 380, Argentine Offer of a Safer Pressurized Water Reactor," Nucl. Eng. Des. 109, 55 ( 1 988). International Atomic Energy Agency, Directory of Nuclear Reactors, Vol. IV, Power Reactors, pp. 157-162, Vienna (1962). International Atomic Energy Agency, Directory of Nuclear Reactors, Vol. IV, Power Reactors, pp. 1 69-1 74, Vienna (1962). E. Critoph, S. Banerjee, F. W. Barclay, D. Hamel, M. S. Milgram, and J. 1. Veeder, "Pros pects for Self-Sufficient Equilibrium Thorium Cycles in CANDU Reactors," Trans. A m . Nucl. Soc., 22, 706 (1975).
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2 1 . P. S. W. Chan and A. R. Dastur, "Fueling Schemes for the Conversion o f Existing CAN DU's from Natural to Enriched Fuel Cycles," Proc. Topical Mtg. Advances in Fuel Manage ment, American Nuclear Society, La Grange Park, IL (1 986). 22. S. R. Hatcher, F. N. McDonnell, J. Griffith and P G. Boczar, "Future Fuel Cycles for CANDU Reactors," Trans. Am. Nucl. Soc., 56 (Supplement), 520 (1988). 23. International Atomic Energy Agency, Directory of Nuclear Reactors, Vol. IV, Power Reactors, pp. 3-8, Vienna (1 962). 24. W. J. Dowis, "Basis of Design for Hanford New Production Reactor (NPR)," HW-SA2981 , Hanford Works (Apr. 1 963). 25. J. A. Wright, "Supercritical Pressure Reactor Technology," Paper No. 65-8, Westinghouse Nuclear Power Seminar (1 965). 26. G. F. Lynch, "SLOWPOKE-Its Application to District Heating," Tra ns. Am. Nucl. Soc 56, (Supplement), 1 73 (1 988). 27. C. A. Goetzman and K. E. Schroeter, "Development of Dedicated Nuclear Heating Plants in the Federal Republic of Germany," Trans. Am. Nucl. Soc., 56 (Supplement), 1 80 (1988). 28. L. S. Tong, "Approaches and Principles of Reactor Safety Evaluation and Improvement," Nucl. Safety, 32, 1 76 (1991 ). 29. C. W. Forsberg and W . J. Reich, "Worldwide Nuclear Power Reactors with Passive and Inherent Safety," Oak Ridge Nat. Lab Report ORN L/TM- 1 1 907, Oak Ridge, TN (1991). 30. C. W. Forsberg and A. Weinberg, "Advanced Reactors, Passive Safety and Acceptance of Nuclear Energy," Ann. Rev. Energy, IS, 1 33 (1 990). 3 1 . S. N. Tower, T. L. Schultz and R. P Vejuk, "Passive and Simplified System Features for the Advanced Westinghouse 600 MWe PWR," Nucl. Eng. Des. 109, 1 47 (1988). 32. P. J. Andrews, S. F. Hall and I. H. Gibson, "SIR Reactor Safety and Decommissioning," British Nucl. Energy Soc. (Apr. 1990). 33. K. Hannerz, "Towards Intrinsically Safe Light-Water Reactors," ORAU/IEA-83-2(M) Rev., Oak Ridge Assoc. Univ. (1 983). 34. S. Glasstone and M. C. Edlund, Elements of Nuclear Reactor Theory, p. 198, D. Van Nostrand Company, Princeton, NJ ( 1 952). 35. S. Glasstone and M. C. Edlund, Elements of Nuclear Reactor Theory, p. 215, D. Van Nostrand Company, Princeton, NJ (1 952). 36. "1000 MW(e) Closed Cycle Water Reactor Study," WCAP-2385, Vol. I, Westinghouse Atomic Power Division (1 963). 37a. B. Rothleder, "Power Reactor Fuel Management Beyond First Cycle Operation," Trans. Am. Nucl. Soc., 14, 89 (1971). 37b. J. R. Secker, F. J. Frank, "Optimal Axial Blanket Design for PWR's," Trans. Am. Nucl. Soc. 45, 10 (1983). 38. C. Bangie, G. Gambier and M . Soldevilla, "An Improvement in the Use of Plutonium in Pressurized Water Reactors," Nucl. Technol., 86, 275 (1989). 39. International Atomic Energy Agency, Directory of Nuclear Reactors, Vol. lV, Power Reactors, pp. 21-32, Vienna (1962). 40. J. K. Pickard, Ed., Nuclear Power Reactors, pp. 77-80, D. Van Nostrand Company, Prince ton, NJ (1957). 41. A. Radkowsky and R. T. Bayard, "Physics of Seed and Blanket Cores," Proc. 2nd U.N. Conf. Peaceful Uses At. Energy, Vol. 1 3, p. 1 28, United Na tions, New York (1958). 42. "Shippingport L WBR Preliminary Safeguards Analysis Report," Bettis Atomic Power Laboratory (1975). 43. A. Radkowsky and Z. Zhager, "Feasibility of a High Gain Light Water Breeder Reactor on the U-Pu Cycle," Proc. Topical Mtg. Advances in Fuel Management, Pinehurst, NC, March 1 986, p. 33, American Nuclear Society, La Grange Park, IL (1986). 44. C. Broeders, M . DaIle-Donne, "Conceptual Design of a (P,U)02 Core with a Tight Fuel Rod Lattice for an Advanced Light Water Reactor," Nucl. Technol., 71, 82 (1985). 45. H. Markl, C. Goetzmann, and H. Moldaschl, "KWV's High Conversion Reactor Concept," Nucl. Technol., 80, 65 (1988). 46. E. Saji, Y. Akeyama, N. Kono, and K. Nambu, "Feasibility Studies on High Conversion Pressurized Water Reactors with Tight Core Configuration," Nucl. Technol. , 80, 18 (1988).
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OF
PWR S
4 7 M. C. Edlund, "High Conversion Ratio Plutonium Recycle in Pressurized Water Reac tors," A mI . Nucl. Energy, 2, 801 (1 975). 48. B. C. Armstrong and R. J. Buechel, "Control Rod Repositioning Considerations in Core Design Analyses," Trans. Am. Nllcl. Soc. 62, 555 (1990). 49. G. N. Hamilton and J. A. Roll, "Power Distribution and Reactivity Measurements in Crit ical Lattices Containing Thimble Control," WCAP-1894, Westinghouse Atomic Power Di vision (1964). 50. T. Morito, L. R. Scherpell, K. J. Dzikowski, R. E. Radcliffe, and D. N. Lucoff, "Topical Report on Power Distribution Control and Load Following Procedures," WCAP-8403, Westinghouse Atomic Power Division (1 974). 5 1 . S. Glasstone and M. C. Edlund, Elements of Nllciear Reactor Theory, p. 266, D. Van Nostrand Company, Princeton, NJ (1952). 52. A. Amouyal, P Benoist, and J. Horowitz, "Nouvelle Methode de Determination du Fac teur d'Utilization Thermique d'une Celle," J. Nucl. Energy, 6, 79 (1957). 53. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, pp. 382-389, Addison-Wesley Pub lishing Company, Inc., Reading, MA (1966). 54. W H. Walker, "Mass Balance Estimate of Energy Released per Fission in Reactors," AECL-3109, Atomic Energy of Canada Ltd. (1968). 55. G. Joanou and J. Dudek, "GAM-I : A Consistent P-l Multigroup Code for Calculation of Fast Neutron Spectra and Multigroup Constants," GA-1 850, General Atomics Co. San Diego, CA (1961). 56. H. Honeck, "THERMOS: A Thermalization Transport Theory Code for Reactor Lattice Calculations," BNL-5826, Brookhaven National Laboratory (1962). 57 W. Cadwell, "PDQ-7 Reference Manua!," WAPD-TM-678, Bettis Atomic Power Labora tory (1 967). 58. A. F. Henry, Nuclear Reactor Analysis, MIT Press, Cambridge, MA (1975). 59. M. K. Butler and J. M. Cook, "One-Dimensional Diffusion Theory," Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, Eds., Gordon and Breach Science Publishers, New York (1 968). 60. P G. Justis and G. W. Smith, "Implementation of Thermal-Hydraulic Feedback in GRPDQ," Trans. Am. Nucl. Soc., 59, 319 (1989). 61. W. T. Sha, S. M. Hendley, and G. H. Minton, "An Integral Calculation of Three Dimensional Nuclear-Thermal-Hydraulic Interaction," Trans. A m . Nucl. Soc., 9, 476 ( 1 966). 62. D. L. Delp, D. Fisher, J. M. Harrema, and M. J. Stedwell, "FLARE: A Three-Dimensional Boiling Water Reactor Simulator," GEAP-4598, General Electric Company (1964). 63. T. Kiguchi and T. Kawai, "A Method for Optimum Determination of Adjustable Para meters in the Boiling Water Reactor Core Simulator Using Operating Data on Flux Dis tribution," Nucl. Technol., 27, 315 (1975). 64. L. Goldstein, F. Nakachi, and A. Veras, "Calculation of Fuel-Cycle Burnup and Power Distribution of Dresden I Reactor with the TRILUX Fuel Management Program," Trans. Am. Nucl. Soc., 10, 300 (1967). 65. K. Chitkara and J. Weisman, "An Equilibrium Approach to Optimal In-Core Fuel Man agement for Pressurized Water Reactors," Nucl. Technol., 24, 33 (1 974). 66. y. A. Chao and C. W. Hu, "A Theory of Fuel Management Via Backward Diffusion Cal culations," Nucl. Sci. Eng., 93, 78 (1986). 67 A. Ahlin and M. Edenius, "CASMO-A Fast Transport Theory Assembly Depletion Code for LWR Analysis," Trans. Am. Nucl. Soc., 26, 604 (1977). 68. M. Edenius, "CASMO-3-A Fuel Assembly Burnup Program," NFA-88148, Studsvik (1988). 69. "Qualification of PHOENIX-P / ANC Nuclear Design System for Pressurized Water Re actor Cores," WCAP-1 1 596, Westinghouse Electric Corp., Pittsburgh, PA ( 1987). 70. R. D. Lawrence, "Progress in Nodal Methods for the Solution of the Neutron Diffusion and Transport Equations," Prog. Nucl. Energy, 17, 271 ( 1 986). G. Greeman, "Recent Advances in Analytic Nodal Methods for Static and Transient Re actor Analysis," Proc. Topl. Mtg. Computational Methods in Nucl. Eng., Williamsburg, VA, p. 3 (1979).
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72. Y. Liu, "ANC-A Westinghouse Advanced Nodal Computer Code," WCAP-l 0996-A, Westinghouse Electric Corp., Pittsburgh, PA (1985). 73. K. S. Smith, "Application of Diffusion Theory Methods to PWR Analysis," Trans. Am. Nucl. Soc., 60, 329 (1989). 74. S. Borreson, "A Simplified Coarse-Mesh Three-Dimensional Diffusion Scheme for Cal culating Gross Power Distribution in a Boiling Water Reactor," Nucl. Sci. Eng., 44, 37 (1971). 75. R. B. Stout and A. H. Robinson, "Determination of Optimum Fuel Loading in Pressurized Water Reactors Using Dynamic Programming," Nucl. Technol., 20, 73 (1 973). 76. Y. A. Chao and J. A. Penkrot, "A Nodal Method for Direct Intra-Node Distribution Cal culator," Proc. Topl. Mtg. Advances in Reactor Physics, Kiamesha Lake, NY, NRC NUREG j CP0034 (1982). 77 Y. A. Chao and J. A. Penkrot, "Diffusive Homogeneity-The Principles of the Superfact Multidimensional Nodal Code SUPERNOVA," Trans. Am. Nucl. Soc. 55, 583 (1 987). 78. S. Glasstone and M. C. Edlund, Elements ofNuclear Reactor Theory, p. 243, D. Van Nostrand Company, Princeton, NJ (1952). 79. A. U. Rehman, G. V Guardalben, A. L. Wright, G. M. Frescura, "Design of Cobalt Ad juster Rods for New Pickering NGS-A Core," Proc. Topical Mtg. Advances In Fuel Man agement, Pinehurst, NC, p. 87, American Nuclear Society, La Grange Park, IL ( 1 986). 80. A. A. Bishop, P Nelson, and E. A. McCabe, Jr., "Thermal and Hydraulic Design of the CVTR Fuel Assembly, CVNA-1 1 5, Westinghouse Atomic Power Division (June 1 962). 8 1 . P. S. W. Chan and A. R. Dastur, "Fueling Schemes for the Conversion of Existing CAN DUs from Natural to Enriched Fuel Cycles," Proc. Topical Mtg. Advances in Fuel Manage ment, Pinehurst, NC, p. 55, American Nuclear Society, La Grange Park, IL (1986). 82. D. Randall and S. St. John, "Xenon Spatial Oscillations," Nucleonics, 16, 3, 82 ( 1 958). 83. R. K. Haling, "Operating Strategy for Maintaining Optimum Power Distribution Through Life," TID-7672, USAEC, Division of Technical Information, Oak Ridge, TN (1963). 84. M. Kitlan and R. W. Miller, "Operational Benefits of Relaxed Power Distribution Control Limits," Trans. Am. Nucl. Soc. 55, 570 (1 987). 85. A. Gautier, A. Darraud, M. Dubourg, C. Dehon and B. Gueselon, "Pressurized Water Reactors: Towards Improved Maneuverability," Framatone, Paris, France (1 984). 86. P. Bacher, "France Operates in the Grey Mode," Nucl. Eng. Int., 30, 12, 26 (1985). 87. T. Morita, B. Carlsen, J. Kutz, "Load Follow Operation with the MSHIM Mode," Trans. Am. Nucl. Soc., 56 (Supplement 2), 73 (1988). 88. W. S. Park and J. M. Christenson, 'The L-G-Bank Control Concept: Improved Method for Load Following," Trans. Am. Nucl. Soc., 62, 35 ( 1 990). 89. S. Glasstone and M. C. Edlund, Elements of Nuclear Reactor Theory, p. 65, D. Van Nostrand Company, Princeton, NJ (1 952). 90. J. F. Perkins and R. W. King, "Energy Release from Decay of Fission Products," Nucl. Sci. Eng., 3, 726 ( 1 958). 91. M. R. Smith, "The Activity of Fission Products of U-235," AEC Report XDC-60-1 - 157, General Electric Company ( 1 959). 92. K. Shure, "Fission Product Decay Energy," WAPD-BT-24, Bettis Technical Review, West inghouse Electric Company (1961). 93. American Nuclear Society Draft Standard 5.1, "Decay Energy Release Rates Following Shutdown of Uranium-Fueled Thermal Reactors" (revised Oct. 1 973) . 94. R. E. Schenter and T . R. England, "Nuclear Data for Calculation of Radioactivity Effects," Trans. Am. Nucl. Soc., 22, 5 1 7 (1 975). 95. Code of Federal Regulations, Chap. 1 0, Part 50, Appendix K, Licensing of Production and Utilization Facilities, U.s. Government Printing Office, Washington, DC. 96. R. E. Schenter and F. Schmittroth, "Radioactive Decay Heat Analysis," Proc. Conf. Neutron Cross Sections and Technology, Washington, D.C. ( 1975). 97 F. E. Obenshain and A. Foderaro, "Energy from Fission Product Decay," WAPD-P-652, Westinghouse Atomic Power Division (1955). 98. "Draft American National Standard for Decay Heat Power in Light Water Reactors," ANSI! ANS-5.1-1992, American Nuclear Society, La Grange Park, IL (1992).
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OF
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99. "American National Standard for Decay Heat Power in Light Water Reactors," ANSI! ANS-5.1-1993, American Nuclear Society, La Grange Park, IL (1993). 1 00. German National Standard, "Decay Heat Power in Nuclear Fuels of Light Water Reac tors," DIN 25463, Beuth, Berlin (1990). 101. Atomic Energy Society of Japan, "Decay Heat Recommended Values and Methods of Use," Reactor Heat Standards Committee (1 990). 1 02. "Nuclear Energy Light Water Reactors-Calculation of the Decay Heat Power in Nuclear Fuels," Draft International Standards ISO/ DIS 1 0645, International Organization for Stan dardization (1 990). 1 03. J. K. Dickens, T. R. England and R. E. Schenter, "Current Status and Proposed Improve ments to ANSI ! ANS-5.1 American National Standard for Decay Hea t Power in Light Water Reactors," Nucl. Safety, 32, 209 (1991). 1 04. A. G. Croff, "ORIGEN2: A Versatile Computer Code for Calculating the Nuclide Com positions and Characteristics of Nuclear Materials," Nucl. Tech nol., 62, 335 (1983). 1 05. T. Rockwell, Reactor Shielding Desigl1 Manual, pp. 429-433, D. Van Nostrand Company, Princeton, NJ ( 1 956). 1 06. Stuart McLain and J. H. Martens, Eds., Reactor Handbook, 2nd ed., Vol. IV, pp. 429-433, Wiley-Interscience Publishers, New York (1 964). 1 07 T. Rockwell, Reactor Shielding Design Manual, pp. 353-358, D. Van Nostrand Company, Princeton, NJ (1956). 1 08. S. Glasstone and A. Sesonske, Nuclear Power Engineering, p. 615, D. Van Nostrand Com pany, Princeton, NJ (1967).
CHAPTER TWO
F uel E lements
The thermal design of a reactor can be considered to begin with the design of the heat generating elements. As its first step, prediction of fuel element behavior in a reactor requires a knowledge of fuel temperatures during steady-state and transient conditions. Therefore, for the fuels of in terest, we first review the properties necessary for temperature estimation and then examine some of the computational methods involved. We then consider the overall performance of reactor fuel elements. 2.1 2.1 .1 2.1 .1 .1
F U E L ELEMENT C HARACTER I ST I CS Fuels and Their Thermal Properties Metallic A lloy Fuels
Radiation damage to metallic fuel elements is caused largely by highly en ergetic fission fragments. The damage mechanisms fall into four categories: 1 . Displacement of individual atoms (the primary knock-on atoms) from the normal lattice 2. Displacement of additional atoms by the primary knock-on atoms to form displacement spikes 3. The presence of fission products as impurities Cavity formation at points of mechanical weakness.
4.
These changes result in both axial growth and radial swelling of the fuel. Pure uranium metal is not a satisfactory fuel for PWRs designed for power production alone, since it exhibits very poor dimensional stability
T H E RMAL A N AL Y S I S OF PWR S
94
under prolonged irradiation and its high corrosion rate would result in rapid fuel element rupture following any cladding failure. Exposure of the failed fuel to 550°F water for a few hours would completely destroy the fuel element. On the other hand, when reactor operation is for both plu tonium production and power, uranium metal is acceptable because its ra diation stability is adequate during the short reactor exposure designed to limit higher isotope production. Coolant temperatures can be significantly reduced so that the consequences of any failures will be less severe. Much of uranium' s irradiation instability is traceable to the anisotropy of its alpha phase, which leads to expansion in the direction of the (010) crystal plane and contraction in the ( 1 00) plane during low-temperature irradiation. A substantial reduction in anisotropic behavior can be obtained by beta heat treating processes using oil quenching. 1 Further improvement in uranium properties can be obtained by adding a sufficient amount of an alloying element so the gamma phase can be retained at low temperatures. The gamma phase has a cubic structure that does not exhibit anisotropic properties. Considerable effort was expended in attempts to develop alloys fully suitable for PWR service. The most promising were binary alloys of ura nium and molybdenum. 2 Molybdenum is capable of delaying the beta alpha transformation when present as a dilute « 5%) alloying addition. In moderate amounts 5%), it can retain the metastable gamma phase at room temperature. Gamma-stabilized alloys were found to have signifi cantly better aqueous corrosion resistance than pure uranium; alloys con taining 10 to 1 3.5% molybdenum appear to exhibit adequate corrosion resistance for power reactor use when covered by cladding. Their dimen sional stability is better than that of unalloyed uranium; however, while considerably improved, swelling remained a problem. 3 Interest was also shown in gamma-phase alloys of uranium, niobium, and zirconium. A ura nium-molybdenum alloy fuel was initially considered for the blanket of the first Shippingport core, but the superior performance of uranium dioxide led to the use of ceramic fuel instead. Uranium-molybdenum alloys have been successfully used as a fuel for pulsed test reactors.4 The fuel for such reactors must be able to withstand intense shock loading and high dynamic stresses. Total burnup tends to be low; fuel swelling is not expected to prove a problem. The gamma-phase U-I0 wt%-Mo alloy has been the most widely used alloy for this purpose.4 Attention has also been given to developing very-low-alloy uranium fu els for a heavy-water-moderated PWR. The greater neutron economy achievable with low-alloy uranium is highly important in such reactor sys tems. Furthermore, the consequences of fuel element failure are less severe. Since heavy water units are pressure tube reactors, refueling is easier and severe element failures will not propagate throughout the reactor as they might in the close-packed lattice of a light water reactor.
(>
95
F U EL E L EM E NTS
The concept employed has been to form alloys using small amounts of suitable metals that tend to stabilize the gamma or beta phase. When trans formation to the alpha phase takes place, the resulting material consists of small, randomly oriented crystals. The alloys have better dimensional sta bility and corrosion resistance than cast alpha-phase uranium. The alloys examined in depth have included U-2 to 2.5 wt%-Zr, U-0.3 wt%-Cr, U-1 .5 wt% -Mo, and U-Fe alloys. s,6 The so-called "adjusted uranium" alloy that contains � 500-ppm iron, 250-ppm aluminum, 600-ppm carbon, and 20-ppm nitrogen has also been studied. An extensive study of the swelling of low alloys containing varying amounts of iron, silicon, aluminum, and molybdenum was conducted dur ing the development of an improved Savannah River reactor fuel.1 Alloys containing 250- to 350-ppm silicon alone, or in conjunction with other al loying elements-especially molybdenum-were most effective for expo sures up to 5000 MWd /tonne. Alloys containing 350-ppm iron, 350- and BOO-ppm aluminum had satisfactory swelling behavior at temperatures be low 400°C and exposures below 1 3,000 MWd/ tonne. While it has been possible to reduce fuel swelling substantially through alloy addition, the problem has not been completely eliminated. Below �640°C, the swelling obtained varies greatly with alloy content; this effect is consistent with the concept that at these temperatures swelling occurs by mechanical processes. The benefits obtained from alloying additions appear to be due to the influence of the additions on mechanical properties. A minimum swelling rate is reached in the 640 to 700°C range. At tempera tures above this, the rates again increase, but are only slightly different for the various alloys? The Canadians investigated the use of U Si (U-3.B wt% Si) for PWR 3 service. While swelling is considerably lower than that of unalloyed ura nium, it is higher than desired. However, the fuel is sufficiently plastic so the addition of a central void appears to offer a way of accommodating a major portion of the swelling. 8 The water corrosion resistance of uranium is significantly improved by the addition of small amounts of molybdenum, zirconium, Nb Zr, or sil icon. Indeed, U3Si has c. corrosion resistance perhaps 500 times that of un alloyed uranium. However, attack of a fuel element of these alloys with a cladding hole leak would be severe enough to require fairly prompt re moval of the element from the reactor. Small aluminum additions, e.g., 1 .5% aluminum, substantially improve the corrosion resistance of U3Si (Ref. 9). A major inhibition to the use of uranium silicide is that U3Si undergoes rapid and gross swelling at fuel temperatures in excess of 900°C (Ref. 9). Such temperatures can be encountered during a loss-of-coolant accident (LOCA). Expected external pressures are not adequate to prevent severe ballooning. Mathews and Swanson9 suggest that such high-temperature swelling may be typical of all metallic fuels with fission gases in irradiation-
+
96
T H E RMAL A N ALYS I S O F PWR S
induced pores and may pose a severe limitation on the use of metallic fuels in reactors where accident temperatures approach the fuel melting point. High-alloy uraniums with more satisfactory properties are available. Such alloys provide a means for dispersing highly enriched uranium in a structure with a sufficiently large surface area to attain the high heat re moval rates required. Uranium-aluminum alloys have been widely used in research reactors and in some production reactors. However, the poor corrosion resistance of aluminum in high-temperature water limits use of these alloys to low temperature applications. The only high-alloy uranium used for power pro ducing PWRs are alloys of uranium and zirconium. Zirconium' s low neutron absorption cross section, high melting point, and corrosion resis tance, and uranium ' s high solubility in zirconium make it a natural alloying material. Zirconium alloys have adequate aqueous corrosion resistance and adequate strength when they contain a small amount of tin. In actual prac tice, uranium is alloyed with Zircaloy-2, which contains 1 .5% tin plus traces of iron, nickel, and chromium. Dimensional stability under irradiation is good. Alloys containing up to 14 wt% U have satisfactorily withstood burn ups of up to 3.3 at. %. Maintenance of dimensional stability seems to require that the fuel not be cycled above the phase transition temperature (=600°C) (Ref. 1 0). An alloy of uranium and Zircaloy was used for the meat of plate type sandwiches in the first Shippingport reactor seeds. The thermal design properties of U-Zr alloys were investigated by Bat telle Memorial Institute. Etherington1 1 tabulates the thermal conductivity of the alloy and pure components as a function of alloy composition and temperature. Density of the alloy at room temperature is approximately a linear function of the atomic percentages of the constituents. 12 Similarly, specific heats can be estimated from that of the components. Further prop erties can be obtained from Ref. 1 3. A summary of significant properties is given in Table The thermal properties of U-Mo alloys have also been extensively inves tigated. Room-temperature thermal conductivity varies from 0.148 W / cmoC for 8 wt% Mo to 0.138 W / cmoC for 12 wt% Mo (Ref. 13). The thermal conductivity of 8 wt% Mo fuel increases linearly with temperature to 0.39 W / cmoC at 790°C (Ref. 14). Thermal expansion is not linear with temper ature, but shows an inflection at 595°C. A total length change of 1 .8% is seen at 790°C (Ref. 14). Additional properties can be obtained from Ref. 15. Properties of the ternary uranium, 7.5 wt% Mo, 2.5 wt% Nb, alloy are reviewed by Weiss. I S Properties of a number of other alloys of interest are available in Refs. 13 and 1 6. Thermal properties for the most important alloys are summarized in Table 2.1. Metallic thorium containing small amounts of enriched uranium has been proposed as a PWR fuel.17 Thorium has a face-centered cubic (fcc) structure and does not exhibit the anisotropic behavior of uranium. The irradiation experiments available (1978) give no indication of instability and
2. 1 .
(200°F) (400°F) (1 200°F)
room temperature-1200°F, orth.
-
Heat capacity [Btu/ (lb OF)]
Thermal expansion [in. /(in. OF)]
"-----
(200°F) (1 000°F) (1 200°F)
--- --
0.0278 0.0410 0.0464
--
20.2 X 10 - 6 for ( 1 00) plane - 5.17 X 1 06 for (010) plane 19.0 x l O - 6 for (001 ) plane Volumetric 34.2 X 10 - 6 7�F to 1 202°F 0.0287 (200°F)
U-1 2 wt% Mo
1 805
7.67 ± 0.28 x l O - 6 (212 to 752°F)
(77°F) ( 149°F)
Zr-14 wt% U
Zr (hex.)
-
-
-
-'--
-
(200°F)
0.0674 0.032 to 0.036 (572 to 752°F)
(68°F) (21 2°F) (392°F) (572°F) (1 292°F) 3.78 X 1 0 - 6 (221 to 626°F) 3.84 X 1 0 - 6 (662 t o 1 022°F)
6.36 6.71 7.12 7.52 1 0.4
7.97 (room temperature) 7.23 X 1 0 - 6 (21 2 to 752°F)
3240
2 102
E Zr (hex.)
Ci
1 6.9 7.16 (room temperature) (room temperature)
U + U3Si2 > 1 706°F bcc E < 1 706°F tet. (E formed by holding cast alloy at 1 472°F)
(95°F) 8.66 (572°F) 1 0 . 1 (1 1 12°F) (1 652°F)
8.0 X 1 0 - 6 ( 1 05 to 930°F)
1 2.7 1 5.6 2 1 .4 27.8
1 5.8 1 7.5 20.25 22.00
Conductivity [Btu/(h ft OF)]
Orth.
2060
(200°F) (600°F) (1 000°F) (1400°F)
U3Si (U-3.8 wt% Si)
1 5.57 ± 0 .02 1 8.3 (room temperature) (room temperature)
U-2 wt% Zr
2070
(bcc)
'"V, 1400°F-mp, body-centered cubic
Ci,
�, 1 200 to 1 238°F, tet.
1 9.05 1 8.87 1 8.33
Uranium
Melting temperature (OF)
Crystal structure
Fuel density (g /cm3)
Fuel
TAB L E 2.1 Thermal Properties of U ranium Al loys
-0 '-.J
m Z -i Vl
$:
.,.,
C m rm r m
98
TH E RMAL A N A L Y S I S OF PWR S
restrained swelling is 1 to 2% for 1 0,000 MWd / tonne exposure at 650°C (Ref. 1 7). The corrosion rate of pure thorium in 300°C water is two orders of magnitude lower than that or uranium. Addition of up to 6% uranium does not appear to affect corrosion rate. A further reduction in corrosion rate is achieved by small additions (0.03 to 0.12 wt%) of carbon. The carbon addition also leads to a more than threefold increase in yield stress. Thorium is a semirefractory metal (melting point of 1 845°C) and has a good thermal conductivity (�0.1 cal ! cm SOC). Hence, fuel temperatures during a LOCA should be far below the melting point and disastrous swell ing would not be expected. However, no direct tests of this appear to be available (1992). Use of thorium metal instead of Th02 in an LWBR is pre dicted to lead to a slight improvement in breeding ratio and a reduction in seed enrichment.1 7 At present (1992) there is essentially no industrial interest in using me tallic thorium or low-alloy metallic fuels for lightwater reactors (LWRs). The technology of ceramic fuels is satisfactory and so well established that there is no real incentive for departure from ceramics for central station power generation. There is interest in using an intermediate metal alloy (62.5% U, 15% Pu, 22.5% Zr) for fast reactor fuel elements. Fuel rods with a large, sodium-filled pellet-to-cladding gap can accommodate significant radiation damage. This alloy shows sufficient thermal expansion so that a sodium-cooled fast reactor with this fuel can survive an unprotected loss of-flow accident without core damage. Such behavior is not achievable with ceramic fuel rods. Based on the renewed fast reactor interest in alloy fuel, Lancaster al . 1 8 have more recently (1992) proposed the use of a uranium metal alloy con taining 10% zirconium as a fuel for water-cooled central station reactors. They suggest that the low centerline temperatures of metallic fuels, which lead to substantially less stored energy, are a significant safety advantage during LOCAs. Lancaster al. believe high burnups can be achieved by using annular pellets. However, it is not at all clear that the corrosion re sistance of this alloy would be sufficient for PWR use. High-alloy fuels are not of interest for central station use because of the high costs associated with the use of the highly enriched uranium they require. However, these fuels would obviously be of interest for naval re actor cores where they can provide a means for obtaining both high specific power and long core life.
et
et
2 . 1 . 1 .2
Dispersion Fuels (Cermets)
An alternative way to provide a structural base with the necessary high surface for highly enriched fuel is to physically disperse the fuel in a me tallic matrix. Dispersions of U02 in aluminum, zirconium alloys, and stain less steels have been investigated, but only stainless-steel and zirconium alloy dispersions have been used in PWRs.
F U E L E L E M E N TS
99
Stainless-steel-U02 dispersions are normally fabricated by powder metallurgy techniques. Uranium dioxide and stainless-steel powder are blended, pressed, and sintered. Final dimensions and further densification are achieved by cold or hot working. The physical and mechanical prop erties of the dispersions are influenced by many variables including U02 particle size, type and amount of U02 , type of stainless steel, geometry of the element with respect to the property being measured, and fabrication tech niques. Reference 1 9 provides a compilation of mechanical properties. Zircaloy-U02 dispersions have been used as a replacement for the U-Zr alloy used in the first seed of the Shippingport reactor. Such dispersions have greater resistance to radiation damage than the alloy, since damage would be confined to the area around the U02 particles rather than being dispersed throughout the alloy. There has been no interest in the use of dispersion fuels for central station cores since the virtual abandonment of the seed-and-blanket concept. How ever, dispersion fuels would clearly be of major importance for naval re actor cores. These fuels could lead to cores with high specific powers and lifetimes that could exceed those obtained from zircaloy-uranium alloy fuels. Normally a dispersion fuel element offers good corrosion resistance re gardless of the resistance of fuel particles. Exposure of small amounts of fuel to the coolant due to cladding failure means only a few particles are exposed. If there is stringering of the fuel particles, corrosion can proceed along the interconnected particles. Since U02 has good corrosion resistance in high-temperature water, the aqueous corrosion resistance of these dis persions is good even when some stringering occurs; however, the radiation damage resistance of such a fuel would be poor. Resistance to radiation damage of stainless U02 dispersions is good, but as with all fuels, radiation damage places some limits on their performance. In a properly fabricated dispersion fuel, the particles are uniformly dis persed throughout the nonfissile matrix that predominates in volume. Fis sion products released from the fuel are then confined to narrow regions around the fuel particles; a continuous web of undamaged matrix remains. Radiation-induced swelling arises because of growth of the U02 particles due to an accumulation of fission products and the partial escape of fission gases from the U02 (Ref. The latter effect is the most significant. A gas filled void is created around the particles; this pressurizes the matrix shell causing it to expand as a thick-walled vessel under pressure. Thus, the matrix swelling that can be allowed will set the maximum burnup. Swelling limits can be determined on the basis of allowable dimensional changes or a maximum strain determined on the basis of the reduced ductility limit of the neutron-embrittled matrix. The specific heat of a dispersion can be obtained by combining those of its constituents linearly in accordance with dispersion composition. Densi-
20).
1 00
TH ERMAL A N A L Y S I S OF PWRS
ties can be obtained in a similar manner providing an allowance is made for the fact that the U02 particles are porous and, therefore, densities of the dispersion range from 92 to 98% theoretical. To obtain the thermal con ductivity of a dispersion of small particles of conductivity, uniformly dispersed in a continuous substance of conductivity Jakob2 1 recommends an equation originally derived by Maxwell and used for the present pur poses by Eucken:
ks'
p /ks)b kd =ks 1 -(11 +-ak (a-1)b"
kp
(2.1)
where
kd = a = 3ks /(2ks++ kp ) b = Vp /(Vs Vp ) Vs = Vp = b�O.5(2.. 1 )
thermal conductivity of dispersion
total volume of continuous substance total volume of distributed particles.
b�O.b,25
Equation was derived for small values o f but i t holds approximately for We suggest that use above should be with caution. Stora 22 suggests that for a mixture in which neither phase completely sur rounds the other, Bruggeman' s equation23 should be used:
ks -kd = ( b ) ( kd -+kp ) ks -2kd -b 2kd kp (2. 1 ) 1
(2.2)
If there is appreciable stringering of the U02 , thermal conductivity is anisotropic and Eq. should not be applied. Under these conditions, Stora 22 suggests that the Fricke equation24 for dispersion of ellipsoidal par ticles can be used. He considers that all particles can be reduced to ellip soids by appropriately varying the relative size of the A , B, and C axes. The anisotropic effect is simulated by changing the size and orientation of the axes of the ellipsoids with respect to the direction of heat flow. Fricke' s equation for a dispersion is
(2.3) where
When axis
x A
[3 -1)++ 1 = (k(kpp/ks)( /kJ -([3 1)
is parallel t o the heat flow, [3
(kp /kJ -1 , 1 + [ (kp / kJ -1 ] (1 -M)
(2.4)
1 01
F U E L E L EM ENTS
and when axis A is perpendicular to the heat flow,
(kp /ks ) - 1
(2.5)
M
Here, is a trigonometric function of the AlB length ratio. By using the X parameter, Stora 22 extended Bruggeman' s equation for mixtures in which neither phase completely surrounds the other to ellipsoidal particles. (2.6) However, his application of this equation to U02-metal mixtures, in which the metal fraction � 0.3, yielded k values that differed from measured val ues by as much as 40% . 2 . 1 . 1 .3
Uranium Dioxide
This is by far the most popular fuel material for LWRs; uranium dioxide, mixed uranium and plutonium dioxides, and mixed thorium and uranium dioxides are the only ceramic fuels that have received serious attention. Although the carbides can be satisfactory from a radiation damage view point, their reaction with water eliminates them from consideration for LWR service. Uranium dioxide has been found to exhibit excellent dimen sional stability to high burnup. It shows appreciable radial cracking,25 but when properly restrained by fuel element cladding, cracking does not lead to dimensional instability. In addition, U02 has a high melting point, is relatively inert chemically, and has good pellet fabrication characteristics. The generation of fission products leads to a swelling of the U02 crystals that is roughly linear with burnup. However, at low burnups, the fuel po rosity can accommodate much of the crystal swelling and external dimen sional changes are small. When the available porosity has been exhausted, swelling rates increase sharply. The burnup at which the swelling rate in creased sharply used to be referred to as the The so-called critical burnup for unrestrained U02 at normal PWR operating conditions is shown in Fig. 2.1 as a function of the void within the fuel. Since the swelling rate also depends on fuel temperature, the critical burnup actually observed depends on the operating power level. An illustration of the effect of cladding restraint is provided by Dayton,S who noted that for platelet-type fuel elements, the increased cladding re straint of 1 /4-in. platelets postponed critical swelling to a burnup over 1 30% that observed for l /2-in.-wide platelets. Sophisticated design calculations now recognize that U02 at high temperatures has some plasticity; defor matio n of the fuel is affected by elastic, plastic, and creep properties of fuel and cladding materials. 26 While Fig. 2.1 was formerly used as a rough guide for determining de sign burnup limits, present design approaches are considerably more so-
critical burnup.
T H E RMAL A NA L Y S I S OF PWRS
1 02
j
60 000
E
50 000
Q) c c
)
o
40 000 � -u
�
e" '2
::; ..... o
30 000
c
:J 20 000 E :J In Cl.
1 0 000
a
L
/V /
V
/
/ /
/
V
I/'
V
�V
/
2
4
6
8
10
Percent Voi d Vol ume a t Temperature Fig. 2 . 1
Critical burnup for rapid swelling of unrestricted U02 [from NIlc/. El1g. Des., 6 , 301 ( 1 967) ] .
phisticated. The achievement of low fuel costs requires burnups in excess of the critical burnup and fuel rods are now designed to accommodate the increased swelling obtained. At high flux irradiation, appreciable U02 sublimation occurs; this leads to the formation of a central void surrounded by U02 of essentially theo retical density. Void migration has been theoretically investigated by De Hales and Horn27 and Nichols,28 who derived equations describing the rate of void migration for central temperatures above 2750°F. At very high power, a molten core is obtained. Although some experi mental data indicate such a core is not deleterious,29 the avoidance of melt ing was generally accepted as a design criterion. The volumetric expansion of U02 on melting can lead to severe swelling of the cladding and, subse quently, to its destruction over a portion of the fuel rod. Failures of this
1 03
F U EL E L EM ENTS
et
nature were observed by Eichenberg et a1.30 Successful operation with gross central melting was satisfactorily accomplished by Lyons al. 31 with hol low pellets that provided the free volume necessary to accommodate U02 expansion on melting. Later in-reactor experiments by Lepescky a1. 32 in dicate satisfactory fuel performance with central melting can be obtained with solid pellets. It has been postulated that under particular conditions of fuel relocation and cracking, molten fuel could come in contact with the cladding, which could be brought to its melting point and burned through. This mechanism may be one of the factors contributing to the 1 966 failure in the Plutonium Recycle Test Reactor (PRTR) rupture test loop, where fuel element failure was followed by burn-through of the surrounding pressure tube. The ques tion of whether central melting is a real limit is probably moot since acci dent analyses now generally lead to the establishment of linear heat ratings considerably below those at which melting occurs. Current technology provides U02 compacts of close to theoretical density by pressed and sintered pellets. As the maximum burnups increased, lower density fuels were used to reduce swelling. This led to renewed interest in vibratory compaction, swaging, or combined compaction and swaging of U02 powder as production processes. However, use of low-density fuels can lead to in-reactor densification. In some cases, this led to cladding col lapse into the gaps in the pellet stack created by the densification. The use of high-density pelletized fuel remains the current (1995) commercial prac tice since high-density pellets mean less risk of fuel densification during irradiation. Interest could conceivably revive in vibratory compaction (sphere-pac fuel) because of the ease with which this process could be adapted to remote fabrication of mixed-oxide (MOX) fuel. Under usual PWR operating conditions, U02 does not react with the coolant and corrosion resistance is good. Therefore, pinhole cladding fail ures do not lead to washout of the fuel element. However, waterlogging of failed elements at low power and subsequently expelling steam at higher power does lead to the escape of some fission products into the coolant. Under some conditions, it is possible for the waterlogging effect to be severe enough to cause fuel element failure. Eichenberg af. 33 postulated that this waterlogging failure could occur if the defect were blocked by a particle of U02 , thus restricting the escape of steam during startup. Only a very few such failures have been observed in a large number of in-pile tests30 with sintered pellets; no such failure of actual PWR sintered-pellet fuel elements in reactor service has yet (1 994) been observed, although defective fuel el ements have been present. Waterlogging seems to be more of a problem with vibratory compacted or swaged fuel elements than with those using sintered pellets. Because of the wide use of U02 as a reactor fuel, its properties have been the subject of numerous investigations. Specific heat can be computed as34
et
et
T H E RMAL A N AL Y S I S OF PWR S
1 04
K, 82 exp(8 / T)
YK E
R
+ 2K2 T + 3 2 2 T ep = T 2 [ exp( 8/T ) - I F
exp ( - EfRT )
(2.7)
where T = temperature, K 8 = 535.285 E = 1 .577 X 1 05 296.7 Kl K2 = 2.43 X 1 0 K 8.745 X 107 } Cp = specific heat at constant pressure, J /kg K 8.3143, J / mol K Y = oxygen-to-metal ratio (2 for stoichiometric U02 ). The theoretical density of U02 at room temperature is 1 0.96 g/ cm} Its thermal expansion can be obtained from the simple polynomial expression shown in Fig. 2.2. The density of U02 just below and just above the melting point is reported as 9.65 and 8.80 g/ cm3 , respectively.35 Christensen36 gives the thermal expansion coefficient of molten U02 as 3.5 X 10 - 5 in. /in. DC. Despite the many investigations of U02 thermal conductivity, consider able scatter in the data remains. Data of a number of investigators for unir radiated U02 are shown in Fig. 2.3, where all have been corrected to 95% theoretical density. The simplest way to correct for density variation is to use the rela tionshi p
R
k95o/c
0.95 = kmeasurcd 1 -a
-
(2.8)
where a is the void fraction of the sample. Godfrey et az. 3 7 and May et al.}R noted that the conductivity of U02 increased as the oxygen-uranium ratio is reduced. This stoichiometric effect can account for some of the data scat ter. Belle et al. 39 show that data scatter is reduced when a revised method is used for correcting the data to a standard porosity. By assuming that a pellet is a dispersion of nonconducting pores in a U02 matrix, they used a modified version of the Maxwell-Eucken relationship, Eq. (2. 1 ), to obtain I -a
kp = + k 1 l3a ( 1 QO% )
(2.9)
where kp is the conductivity of the porous sample in question and 13 is a constant that depends on the material. A reduced data spread can be ob tained by using 13 = 0.5 for fuels of 90% theoretical density and above and 13 = 0.7 for fuels of lower density. Stora 22 notes that 13 corresponds to X in Fricke' s equation, Eq. (2.3), if a pellet is considered to consist of a dispersion of pores in U02 . A 13 value
1 05
F U E L E L E M E NTS
8 .-------,--'--�1--�
/
Burdi c k and Pa rker o Lam berson and Ha ndwerk 6 Be l l and Mak i n
o
7 r-
6-
'#.. 5 c
.gc
o Murray a n d Thackray , U 0 2 . 1 3 • Mu rray and T ha ckray, U 0 2 . 0 • C onwa y e t a l . • C hr i stensen • Hock and Mom i n
-
-
r-
o c.. x UJ
-
-
-
-
I
2000
L
2500
3000
Temperature , ° C Fig.
2.2
Comparison of U02 thermal expansion data with prediction [from Ref. 53] .
of 0.5 corresponds to spherical pores, while a larger value corresponds to ellipsoidal pores. Thermal conductivity data below 2500°F generally follow the expected behavior for lattice conduction. In 1 961, on the basis of postirradiation ex aminations, Bates40 postulated an increase in conductivity at higher tem peratures due to internal thermal radiation. This seems to be borne out by
TH E RMAL A N A L YS I S OF PWRS
1 06
1 0 �----�--.---,--,--r Fay l and Hansen (!J
�
� I E
�
8
y
.-
3'
'J.
... 6 � .0-
o ::l -u
g 4
u
E
o (!)
.:: 2
�
Z
Z
Z
Z
G o ldsm i th and Doug las C raeynest
Stora et a I . H e tz l er
G odfrey
2
H edge
4-
Bates
(!)
Rei sw i g
K i ng ery
Z Z
Z z
O L-----�----�---J 2800 2400 1 600 1 200 2000 400 800 o Temperature, ° C Fig.
2.3
Thermal conductivity o f unirradiated U02 [ from Ref. 53].
the more recent data of Godfrey et al. 37 and Nishijima 4 1 for unirradiated U02 · Recent advances in materials science have led to a considerably im proved understanding of U02 thermal conductivity. It is now recognized that U02 conductivity is due to two mechanisms. At low temperatures, heat is primarily conducted by phonons (quantum mechanical description of vibration of atoms in crystal) . At high temperatures, heat conduction is primarily via small polarons (quantum mechanical description of electron and hole motion). Since the phonon contribution decreases with increasing temperature while the polaron contribution increases as the temperature increases, the thermal conductivity curve shows a minimum value in the vicinity of 2000 K. Harding and Martin4 2 recommend on the basis of theo retical considerations and electrical conductivity measurements at high tem perature that the thermal conductivity of 1 00% dense U02 between 773 and 3120 K be computed by: k = (0.037S + 2.16S X I0 - 4 T ) - 1 + (4.71 S X 1 09 T - 2 )
X exp[ - 16361 IT]
(2.1 0)
where k is the thermal conductivity of 1 00% dense U02 (W 1m K) and T is the temperature in Kelvin. The first term in Eq. (2.10), which has the form
1 07
F U E L E L EM E N TS
(A - 1 , represents the phonon contribution; the second term represents the polaron contribution. Harding and Martin estimate that the uncertainty at K and increases to about ± at K. in is about ± A somewhat earlier and more elaborate expression for thermal con is also based on phonon and polaron contri duc tivity by Hagrman butions. It has the advantage of including correction factors for the effects of variation in the oxygen-to-metal ratio, theoretical density, plutonium content, and thermal expansion. In addition, it may be used at temperatures below K. Hagrman recommend:
+ BY)
k
7% 1800 et al.34
15%U02 3120
773 et al. k = [ 1 + (6.5 - ��;469T' ] [ + BT'�(l + 3Eth ) ] + 5.2997 X 1 O-3T [ exp ( - 13�58 ) ] {1 + [ O. 1 69( 13�58 ) + 2 r } (2. 1 1) )(a)
(A
where
k
thermal conductivity, W 1 m K (fraction of theoretical density) void fraction = = phonon contribution to the specific heat at constant volume, J /kg K. The first term of Eq. is used for this factor. = linear strain caused by thermal expansion when temperature is > K, unitless. = fuel temperature, K = fuel temperature if < K. [For temperatures > K, the porosity factor, first bracket on right-hand side of Eq. equals For temperatures in the range of to K, the factor is found by interpolation.] = fuel temperature if < K. For temperatures > K, is equal to K, and for temperatures in the range to K, is found by interpolation. = a factor proportional to the point of defect contribution to the phonon mean free path, m·s/kg·K. The correlation used for this factor is ratio) l . = a factor proportional to the phonon-phonon scattering contribution to the phonon mean free path, m·s /kg·K. The correlation used for this factor is weight fraction Pu02 in fuel).
=
1
a =
Cv £ th
T T'
Til A
B
(2.7)
300
1364
1.0. 1800 2050 Ti l 0.339+11. 1 XI (2.0 - 0IM
1834(2.11), 1364 1384 23001800 Ti2300 l
0.06867 X (1 + 0.6238 X The terms prior to the plus sign in Eq. (2.11) represent the phonon contri bution to thermal conductivity, while the terms following the plus sign represent the polaron (electron-hole) contribution. The expression is valid only in the range of 90 to 100% of theoretical density. When the fuel is molten, the terms preceding the plus sign are neglected.
T H E RMAL A N A L YS I S OF PWRS
1 08
Note that any departures from a stoichiometric V02 results in a reduc tion of the conductivity.43 This is accounted for in Eq. (2. 1 1 ) by the variation of A with oxygen-to-metal ratio. Considerably more uncertainty is attached to the thermal conductivity of V02 after irradiation than to unirradiated V02 conductivity. In 1 959, Runnals44 noted a decrease in conductivity at low temperatures after an irradiation of 9 X 1 017 n/ cm2 and no significant change afterward. Other investigations generally noted qualitatively similar behavior; Fig. com pares the suggested curve for unirradiated conductivity with the available data for irradiated U02 . It seems probable that there is some decrease in conductivity below 15000P and little change between 2000 and 3000°F. In 1 967, Belle al. 39 surveyed all the available data and concluded that conductivity was a function of both burnup and temperature. They pro posed a correlation that included a significant burnup effect. Subsequently, Stora et a1.45 argued that the apparent irradiation-induced reduction in ther-
2.4
et
L _B_
.- -
Tem p erature,
Fig.
2.4
0
F
Irradiated U02 thermal conductivity and out-of-pile data [from Nucl. Eng. Des., 6, 301 (1967) ] . Notation: A = curve based on unirradiated data B = from Ref. 40, postirradiation measurements C = from Bain and Robertson, J. Nucl. Mater., I, 1 09 ( 1959), postirradiation data D = Dayton and Tipton, BMI-1448 (Rev.), Battelle Memorial Institute (1960), postirradiation data (----heating, cooling) E = from Dayton and Tipton, BMI-1448 (Rev.), Battelle Memorial Institute (1960), postirradiation data (6. first-cycle heating, 0 second-cycle heating, 0 third-cycle cooling) F = Lyons et ai., Trans. Am. Nucl. Soc. 7, No. 1 , 1 06 (1 964), �ostirradiation data G = from Robertson et al., J. Nucl. Mater., 7, 225 (1962), pH o�c kdt = 97 W / cm.
1 09
F U E L E L E M E NTS
mal conductivity is, in reality, due to thermal stress cracking. They ob served that the magnitude of conductivity reduction increased as the gap between cladding and pellet increased. When a zero gap existed, essentially no conductivity reduction was observed. They suggested that when crack ing occurs, the pellet segments move apart to the degree allowed by the available fuel space; thus, the small gaps introduced into the pellet reduce the effective conductivity of the fuel. Since the amount of cracking gradu ally increases up to some maximum with operation, the effect could be irradiation related. MacDonald and Weisman46 provided a quantitative analysis of this effect (Sec. 2.2.5). There is now general agreement that most of the observed changes in apparent V02 conductivity with irradiation are due to cracking of the V02 and subsequent healing of the cracks with V02 swelling. However, current design approaches generally note that some changes do occur due to the buildup of fission products and plutonium in the fuel and the changes in fuel porosity with irradiation. The effect of plutonium content is considered in Eq. (2. 1 1 ) by its effect on the parameter B and by inclusion of the phonon contributions to specific heat which vary with fuel composition. The thermal conductivity decreases somewhat as Pu content increases. Perhaps the most important effect of high burnup on the conductivity of uncracked V02 is the introduction of fission products in the lattice.* This results in a reduction of the phonon contribution to thermal conductivity. For thermal conductivity equations such as Eq. (2. 1 1 ) where the phonon contribution has the form (A + BD - 1 , it has been proposed that A be written as [1 (burnup)] . Unfortunately, there are now (1 992) no values for in the open literature. Krammen and Freeburn48 propose a more readily used burnup correc tion. They simply multiply the conductivity of unirradiated U02 by a cor rection factor. They propose
(C,J,
+C
C
for 0 � Bu � 2 x 104 for 2 X 104 < Bu � 3 X 104 for 3 X 104 < BUB
ko
(2. 12)
kr
where Bu is the average burnup at given axial location (MWd /MTU) and are the unirradiated and irradiated thermal conductivity, and resp ectively. Verbeek and Hoppe47 recommend that, when computing the thermal conductivity of irradiated U02 , the actual porosity of the irradiated fuel ""Bagger, Mogensen, and Walker ( J . Nucl. Mater 211, 1 1 (1994)] find that there is also a marked decrease in the thermal conductivity of the fuel at the outer rim of the pellet. At burnups above about 40 GWd / tonne U, the microstructure of the fuel in the outer 2% of the pellet radius changes due to the presence of a high concentration of small gas-filled pores. This reduces the rim-region thermal conductivity to less than 1 W 1m K.
TH E RMAL A N AL Y S I S OF PWRS
110
matrix be used. Some designers suggest that the effect of irradiation on fuel matrix porosity be combined with the effect of fuel cracking. Krammen and Freeburn48 suggest that both of these effects be accounted for through a porosity correction that has the form:
k = ko O - O . 97 1 a - 6 . 06a2 )
(2.13)
where k is the thermal conductivity of uncracked U02 and a = ap + a relo ; ap is the fuel matrix porosity (void fraction) allowing for original fuel porosity and increase in fuel matrix porosity due to fission gas accumulation (gas eous component of fuel swelling); and a relo is a user-specified fraction of the relocation crack volume addition to fuel porosity. Early studies of the melting point of U02 by Christensen49 indicated that the melting point of U02 is a function of the total irradiation received by the fuel (see Fig. 2.5). Christensen49 found an initial increase in the melting point and then a gradual decrease with burnup. Later studiesSo have con firmed the decrease in melting point with extended burnup but there is some indication that the initial increase at low burnups may be spurious. When less than the stoichiometric amount of oxygen is present, the melt ing point is reduced. The data given by Guinet al.S1 show that U0 1 . 8
et
5600
5400 u.. 0
co. 5 200
·0
Cl..
en c
..
�
Q)
5000
4800
4600
a
4
8
27 000 Fig. 2.5
16 12 Fi ssi on/cc 1 0- 20 50 000 Burnup, MWd/tonne
Melting point of U02
as a
20
24
80 000
28
1 00 000
function of burnup [from Ref. 49].
F U E L E L E M E N TS
begins to melt at 2690°C in contrast to 2750°C for pure V02 . Komatsu et al .so have correlated the melting point of plutonium and uranium oxide as a indicate function that of plutonium content,at metal-to-oxygen ratio, and They the temperature which melting begins, urnup. bmixtures can be correIa ted by [ 1 1 -..128388(2 O.Ol1M)l 1 y2 ] T (2. 1 4) +0 11Y where = melting point of pure unirradiated V02 Y = X + O.016Bu Bu burnup, at. % = mole fraction of Pu in mixture 1M oxygen-to-metal mole ratio. A brief summary of V02 properties is provided in Table 2. 1 1 and exten 2 MacDonald and Thompson,s3 and sive information is provided by Belle, s 34 Hagrman et a1. xed thorium and uranium dioxides wereSubsequent produced forinterest use as acentered reactor fuelMiwith the spectral shift control concept. Tm ,
T
-a
-
I1I( U02 )
171 -
Tm(uo ) 2
=
a
x
=
TAB L E 2 . 1 1 Thermal Properties o f Ceramic Fuels
U02
U02-80 at.% * Pu02-20 at.%
Th02
10.97
1 1 .08
1 0.01
Melting temperature (OF)
4980 (see Fig. 2.5)
5036
5970
Thermal conductivity [Btu / (h ft OF)]
2.54 (930°F) 1 .04 (3630°F) (see Fig. 2.3)
Thermal expansion (in. l in. OF) Heat capacity (Btu / lb OF)
6. 1 2 x 1 0 - 6 (75 to 5070°F) (see Fig. 2.2) 0.057 (90°F) 0.076 (l350°F) 0.090 (31 50°F) 0. 1 2 (4050°F)
Fuel Fuel density (g/cm3)
2.02 1 .04
(930°F) (3630°F)
7.29 5.34 3.59 2.68 2.07 1 .68
(200°F) (400°F) (700°F) ( lOOO°F) ( l400°F) (2400°F)
0.055 0.070 0.078 0.082
(90°F) (l 350°F) (31 50°F) (4050°F)
6.12 x 1 0 - 6 (75 to 5035°F) 0.057 0.075 0.088 0.1 1
(90°F) (l 350°F) (3150°F) (4050°F)
*The 20% Pu02 mixture of Table 2.II is typical of that required for fast reactor fuels. LWR fuels would be in the range of 3 to 5% Pu02 and their properties would be very close to those of U02 since the addition of even 20% Pu02 causes little change in properties.
T H E RMAL A N A L YS I S OF PWR S
TA BLE 2 . 1 1 1 Constants (Ref. 3 9 ) for Thermal Cond uctivity Correlation Material
All
Th02
0
Th02 + 1 0 wt% U02
0.0225
Bil
2.21 x l O - 4 1 . 78 X 10
around the use of Th02 as a blanket in a seed-and-blanket thermal reactor. More recently, there has been some discussion of the use of mixed Th02-U02 fuel in CANDU reactors.54 The behavior of Th02 under irradiation is very similar to that of U02 . Thermal conductivity is affected by fuel porosity and thermal stress crack ing in the same manner as U02 . Belle et aJ.39 were able to correlate the conductivity of Th02 and mixed Th02 U02 by using almost the same polynomial as they used for U02 . For unirradiated Th02 and Th02 U02 they recommend
+
+ (2. 1 5)
- a )[Ao+BoT+(6.23/T)+(49.S2/T2 )] k= ( 1I+�a where
= thermal conductivity, Btu/ (h ft OF) T oR � 0.lower 5 fordensities fuels of 90% theoretical density and above; � = 0. 7 for a = fuel void fraction. The constants Ao and Bo are given in Table 2. I II. thermal conductivities of Th02-Pu02 mixtures containing 1. 1 to 2.below 7In-pile wt% Pu02 were measured by Jeffs,55 who observed conductivities �25% those obtained by laboratory measurements of pure Th0 . Within k
2 the scatter of the data, he saw no difference between the compositions tested. Godfrey et a1.56 measured the specific heat of Th02 over a wide range. These and other thermal properties are summarized in Table
2.II.
2 . 1 . 1 .4
Mixed Plutonium and Uranium Oxides
Interest in U02-PU02 fuels originally stemmed from their application to fast breeder technology. Renewed interest in the behavior of MOX fuels arose with their use in some European PWRs. As expected on the basis of the similarity of the two compounds, there is little difference in the physical behavior of mixed oxides and U02 .* In view of the small amount of Pu02 *Although Pu02 and U02 are similar physically, there are significant neutronic differences that require some modification in core design (see Sec. 1 .2.2.2).
F U E L E L E M E NTS
present inleadPWRto fuel « 7%), use of U02 properties for those of the mixture does not maj o r errors. However, properties of Pu0can2 bearemade. now avail ableTheandspecific more heat accurate estimates of mixture properties correlatedareby substituted the same expression 2 may beconstants 342 used for U02 [Eq. (2.7)] ifofthePu0following K} =347.4 (J/kg K) K2=3.95X10-4 (J/kg K ) K3 = 3.86 X 107 (J Ikg) e =571.0 (K) E=1.967X10-s (J /mol) For MOX fuel, Hagrman et af.34 recommend (2. 1 6) Cp ) uo2 and ( Cp ) PU02 are the specific heats of U02 and Pu02 , respec where (and tively, the weight fraction of Pu02 . The specific heat of molten 34 U0Equation isis given as 503 J Ikg K. 2 or PU0F2(2.11) forconductivity U02 thermalof MOX conductivity contains provision forequa ob taining the thermal fuel. The parameter in this tion isasa function of plutonium content. Further, thefirstfunction Cv would be taken the composition weighted average of the term of theoxides specificis heat equation. The effect of the oxygen-to-metal ratio on mixed alsoTheincluded in expansion Eq. (2. 1 1). of Pu02 for ODC < T < T may be obtained thermal froms3 ilLT = -3.974 X 10-4 8.496 X 10- 6 T + 2. 1 513 X 10-9T2 +3.714 X 10- 1 6 T3 (2. 1 7) T is the temperature (DC). The thermal expansion of MOX fuel can bethewhere estimated by aasweighted average ofheattheinexpansion of U02 and Pu02 in same fashion used for specific Eq. (2. 1 6). ir When theoneplutonium is introduced into thedistribution U02 matrixofviathe neutron radiation, would expect a fairly uniform plutonium incenter). the U0The2 matrix (with properapproaches allowance forindicated flux depression in theclearly fuel property averaging earlier should apply. However,distribution when MOXof fuel is fabricated, the resultingscale. pelletsThehavemia heterogenous plutonium on a microscopic crparticles ostructureembedded of such pellets is characterized by isolated highly Pu enriched in a matrix of U0 (Ref. 57). After irradiation, 2 regions of the pellet but a this microstructure is preserved in the cooler ho mogeneous structure is seen in the high-temperature regions. However, autoradiography shows that the Pu is still concentrated at local spots. One expectarethisslightly Inicroscale to lead to aU0situation Puwouldparticles hottersegregation than the surrounding 2 matrix.where the B
melt !
+
TH E RMAL A N A L YS I S OF PWRS
1 14
In theviewfuelof astheasegregation of Pu0 MOX fuels, it might bea preferable to 2purpose treat dispersion for the of computing thermal con ductivity ofS8 themeasured mixture.the melting points of U02-PU02 mixtures and ob Epstein served melting pointsbinary of 2840pair20°Cwithout for U0a 2maximum and 390 or20°Cminimum for Pu0in2 . His data show a typical the liquidus or solidus curves and, for small additions of Pu0 , his results can 2 beatomclosely approximated by assuming a linearanddecrease inobserved melting point per percent of Pu0 addition. Krankota Craig59 that the00 2 melting point of mixed oxides decreases slightly with burnup. After 85, 0 MWd/tonne burnup, the melting point forof (PUO SUO2.7)0S)02 2anddecreased 50°C. .2PUO. Significant thermal design properties (UO. U0 are 2 8 can be estimatedsumby marized in Table Properties of other mixtures interpolation. More accurate values of thermal conductivity, specificin heat, and thermal expansion are available from the equations given this section. ±
2
±
2.1l.
2.1.2 2.1.2.1
Fuel Element Cladding and Assembly Designs Cladding Properties
Although theexposure most ofcommon PWR fuels have goodwould corrosion resistance, continued any of these to the coolant result in coolant activity levels far beyond those consistent with direct maintenance. Thereof fore, it is the universal practice to clad various fuels by a thin layer corrosion-resistant metal. Ideally, the cladding should be byinertradiation, to the coolant, have a high strength and ductility that are unaffected have a low-neutron-absorption cross section, and one be fabricated easilysome and economically. Since no such material is known, must sacrifice attributes toPWRs gain those believed most important. Modern use stainless-steel-clad or Zircaloy-alloy-clad fueltoexclu sively. Low neutron cross section, excellent corrosion resistance lowfor temperature water, and low cost made aluminum an early candidate this use. Unfortunately,strength commonoraluminum alloys do innothigh-temperature have the neces sary high-temperature corrosion resistance water to allow them toinvestigations be considered.on Argonne National Laboratory (ANL) conducted extensive various aluminum alloys containing small amountsAlthough of nickelthese (i.e., alloys 1 %) and lesser amounts of iron and other constituents. have significantly reduced 6corrosion rates,Stainless ANL (Austenitic) found the rates are still excessive at PWR conditions. o high . Wi t h excellent corrosion resistance and Steel strength, the 300-series stainless steels possess many ofaffected the properties of an ideal cladding. Although their properties are slightly by radiation (yield pointis increases decreases appreciably), radiation resistance good, andandtheyductility are readily fabricated withouttheirspecial tech-
F U E L E L E M E NTS
niques.requires Their major drawback is a relatively highTheseneutron cross section, which an increase in initial enrichment. alloys are not suit ableStainless for reactors where natural uranium is the fuel. steelcracking is knownin high-temperature to be subject to stress-assisted corrosion and stress corrosion water when oxygen and hal 1 reported stress-assisted corrosion failures ogens are present. Duncan et al. 6 ofcracking this nature in thereported cold-worked stainless cladding ofofBWR fuel. Nooxygen such has been in PWRs, perhaps because the lower content ofwhere the coolant. However, it presents a severe problem in superheatin systems, experience has been poor, and would be troublesome designing a supercritical PWR.for a particular case depends on the fabrication The stainless alloy chosen techniquesdesign used.properties. All of theTypical 300-series alloys have veryinnearly the same thermal values are presented Tabl e 2.1V cladding stress levels reached irradiation depend strongly theThethermal expansion and creep of theduring cladding, and highly accurate valueson TAB L E 2 . 1 V Properties o f Cladding Materials
Material
Zircaloy-4
347 Stainless Steel
Composition
1 .3 to 1 .5 % tin 0.2 to 0.22% iron 0.1 % chromium 0.007% max nickel balance zirconium
17 to 19% chromium 9 to 1 2% nickel 0.8% columbium 0.2% max manganese 0.08% max carbon 1 % max silicon
Density"" (g/ cm3)
6.57 (room temperature)
8.03 (room temperature)
Melting point
33200P
2550 to 26000P
Thermal conductivity [Btu / (h ft °P)]
7.71 8.23 8.76 9.59 1 0.43 1 1 .9
(1 000P) (2000P) (4000P) (6000P) (8000P) (lOOOOP) 9.05 X 1 0 - 6 (68 9.25 X 10 - b (68 9.55 X 10 - 6 (68 9.8 X 1 0 - 6 (68 1 0.0 X 1 0 - 6 (68 1 0.25 X 10 - b (68
Mean coefficient"" of thermal expansion (in. / in. OP)
Heat capacity'" [Btu/ (lb OP)]
(1 000P) (2000P) (4000P) (6000P) (8000P) (12000P) 3.21 X 1 0 - 6 (25 to 800°C) (rolling direction)
5.17 X 1 0 - 6 (25 to 800°C) (transverse direction) 0.073 0.076 0.079 0.081 0.083 0.085
(2000P) (4000P) (600°F) (800°F) (lOOOOP) (1 2000P)
8.6 9.0 9.8 10.6 1 1 .5 12.4
to to to to to to
0.12 (32 to 212°P)
""No distinction is made between properties of Zircaloy-2 and Zircaloy-4
1 000P) 2000P) 4000P) 6000P) 8000P) 1000°F)
T H E RMAL A N A L Y S I S OF PWR S
1 16
ofthermal these expansion parametersofare304desired. stainlessSmith as and Was62 give the circumferential LlO = -4.75 10- 4 1.90 10-sT (2. 1 8) where Ll 0/00ofisPWR the thermal strain300andseriesT is(austenitic) the temperature DC . At the temperatures operation, stainless steel will not creep if not subj e cted to fast neutron irradiation. However, in the pres ence ofcansuchbe irradiation, significant creeporiginally can be observed. This irradiation creep correlated by an equation due to McElroy aI.63 2 For 304 stainless steel etunder and slightly modifi e d by Smi t h and Was. 6 uniaxial tensile test conditions, = 8. 70 10- 7 uT{1 -exp[t(2. 1 7 X 102o)]} (2. 1 9) 36. 1 2 10 - expO.44 -O.0027T) where = tensile strain (length/length) 1 MeV), n/ cm2 = mean neutron energy of fast neutrons, MeV = tensile stress, MPa T = temperature, DC. Note that, underobserved PWR conditions, stainless-steel creep isdescriptions only a smallof fraction of that with Zircaloy. More elaborate stainless-steel creep thatalongcombine stress-related (so-called "thermal creep") 64 However, and irradiation creep, with transient creep, are available. while thesenotarerequired neededforforPWR fast analyses. reactor design, this level of complexity is generally InZircaloy the usualwouldPWRrequire configuration, use of stainless-steel cladding instead ofmately that the enrichment be increased by approxi 1 %.PWRs Because of the(993) substantial cost increasecladding. this would occasion, only four currently use stainless-steel This situation could be changed by the development of tight-lattice cores where the epfor ithermal neutron spectrum results in a much lower economic penalty stainless steel. Hafnium-free zirconium has and a very4.5 forlow-absorption cross section (0. 1 8 b compared to 2. 4 3 b for iron nickel) andof good resistance to water corrosion at high temperatures. The addition small of tinofandextensive iron significantly the strength of zir conium.amounts As a result investigationsimproves at the Bettis Atomic Power X
cT
+
Ct
E
=
uT
Zircon ium A lloys.
+
X
Do
X
2
8Eu T
X
(
)
F U E L E L E M E NT S
Laboratory, three alloysresistance-Zircaloy-2, have been developed-3, and that-4.have the required strength and corrosion Zircaloy has been reactors almost universally used power as the production. fuel elementZircaloy-4 cladding in(1.3light-water-cooled built for central % used tin, 0.for22%PWRiron,fuel0.10% chromium, balance zirconium)or -3.hasThisgenerally been cladding rather than Zircaloy-2 is due toHowever, the greater resistance to hydrogen embrittlement shown by Zircaloy-4. the Russian VVER reactors alloy have containing obtained generally satisfactory cladding behavior using a zirconium 1 % Nb. Fuel assem blies designed for very high burnup tend to be fabricated using newer proprietary alloys, such as Zirlo. These show significantly lower corrosion ratesAlthough after long-term exposure tozirconium the reactoris coolant. the melting point of highthis(1852°C), a phasepurechange atconium 862°Cgoes affectsfromthe amechanical properties. At temperature, ziris close-packed hexagonal structure, to one that body-centered cubicZircaloy (bce). Zircaloy, however, transforms overcompletes a rangetheof temperatures. Pure begins to transform at 821°C and transformation at with 960°Cincreasing (Ref. 34).oxygen Transformation temperatures are signif icantly increased concentration. At an oxygen con centration of 2. 5 wt%, the transformation begins at 1283°C and is completed at 2026°Csituations (Ref. 34). This effect of oxygen isoperating generallyconditions, significant only during accident since, under typical the oxygen concentration in Zircaloyis inistheonly phase. about 0. 1 2 wt%. At operating conditions, the Zircaloy cladding Zircaloymusttubing or plate is morecontrolled. expensiveOxygen than stainless steel, andfrom fab rication be more carefully must be excluded all welds;enclosure. weldingSince must zirconium be done inalloys an inertcreepatmosphere, in low a glostresses ve box orat similar at relatively PWR temperatures, allowancecreep mustcanbebemade in the since mechanical design of the fuel elements. However, beneficial it acts to relieve anyZirconium high-cladding stress. and its alloys2 react significantly with release steam at(6.high temper atures [Zr+2H (g)]. The energy 4 5 X 106 J/kg 0(g)-7Zr0 +2H 2 2 Zr)rise due to this reactiontemperatures must be considered when Ataccidents that canabove give to high-cladding are analyzed. temperatures 500°C,thethemetallic hydrogen produced andreducing the oxygen of the ofoxidethe layer diffuse into phase, thereby the ductility metal. Stain less-steel with steam. cladding at high temperatures also reacts chemically However, thereexcesstheofreaction rate does not become appreciable until tem peratures in 1000°C are reached. If interfacial temperatures rise above 675°C, Zircaloy cladding can react with U0 . Since such temperatures are < 400°C in normal operation, the 2 to be considered only for accident situations. reaction needs ex,
ex
TH E RMAL A N A L Y S I S OF PWRS
1 18
operating temperatures, zirconium reacts conditions, slowly withoxygen water tofromAtformthenormal zirconium oxide and hydrogen. At operating oxidediffuses layer through does notthediffuse intolayertheintometal, but hydrogen pro duced partly oxide the metal. The quantity of hydrogen (given as athefraction of thefraction. total hydrogen formed) that diffuses into the metal is called "pickup " Zircaloy-4 has a pickup frac tionIn-reactor betweencorrosion 0.005 and 0.2 (Ref. 65). ofcoolant Zircaloyconditions. is significantly increasedunderby fast neutron flux under oxygenated However, low-oxygen conditions, fast generally neutron fluxholdsincreases the corrosion rate onlyoverpressure slightly. Thisin latter situation in a PWR since hydrogen the pressurizer inhibits radiolysis. In the Saxton core,66 weight gains on the order of 200 mg/ dm2 were observed after 500 full-power days at 339°C. Present modeling of Zircaloy-4 corrosion generally considers it Institute to occur inprovides two stages. The model described by the Electric Power Research an initial nonlinear(post-transition). time dependence followed 67 The(pretransition) bythe amodel linearforare:time dependence equations comprising (2.20)
Thickness at transition: St exp( / RT] -ET[ ) where dS/dt corrosion rate, J.Lm / day S = oxide layer thickness, !-Lm TJ = metal-to-oxide interface temperature, 2 fast neutron flux, (E> MeV), n/ (cm s) The constant values are listed below: =D
-
Q3
(2.21) (2.22)
=
K
Constant A
Ql
Co
M p
1
Value 6.3 X 109 /-Lm) / day 32,289 cal! mol 8.04 X 107 /-Lm/ day 2 1 .91 X 1 0 - 15 cm sin 0.24
Constant
Q2 0
Q3 U E R
Value 27,354 cals/mol 2.14 x 1 07 /-Lm 1 0,763 cals/ mol 2.38 x l Oti /-Lm/ day 2 1.17x 10- K - 1 1 .98 cal / (mol K)
Calculation ofdropthe across metal-oxide interface temperature should allow for the temperature the oxide ( k o d = 0.0168 W / cm K). xi e 68 that external oxidation of the cladding It hasnotgenerally been established should exceed about 100 !-L m . Tests indicate that this limit is reached at about 50 GWd/tonne burnup with Zircaloy-4. This burnup can be exceeded
F U E L E L E M E N TS
1 19
y the1use of newer zirconium alloy claddings. The Zirlo alloy, which con btains % Sn and 1 % Nb, has in-pile corrosion rates 30 to 50% below Zircaloy-4. 69 Alternatively, a duplex cladding having an outermaylayerbe(�100 iJ-Them) low-tin of Zircaloy wi t h extra-low Sn (ELS) bonded to Zircaloy-4 used. outer layer is more resistant to corrosion than the standard Zircaloy-4 Since thewould outer belayeressentially is so thin,thosethe ofmechanical prop ertiesTheoftotalthecladding. duplex cladding Zircaloy-4. corrosion can be related to the average hydrogen concentration, CH, in Zircaloy by using (2.23) where = pickup fraction = total weight gain, mg/ dm2 c = cladding wall thickness, cm = zirconium density, g/ cm3 Hydriding of Zircaloy cladding can also originate from the internal clad ding surface.canInlead the presence oforwater contamination of U0surface 2 pellets,dueinternal hydriding to bulges blisters on the cladding to the 13% volume change in the Zr�ZrH1. 6 reaction. Internal hydriding has been one ofseems the most persistent sources of fuel rod failure. Thetheattack is localized and to occur at local hot spots or regions where zirconium oxide fiinlmthehasgasbeeninsidedamaged. To avoid such attack, concentration of water vapor a fuelspecify rod must be water kept below 2 mg/of U0 cm32 (pellets Ref. 70).beMan ufacturers generally that the contents held atin apellet few ppm so this critical concentration does not arise. Hydrogen trapped pores is sometimes removed bycangetters. react chemically withlocally zirconium. Fission products released from U0 2 Iodine-induced stress corrosion cracking can occur when high 1 The possibility of such brittle stresses arise, such as during power ramps. 7 cracking means thatfuelconsiderable care must be exercised in thestresses rate atcanwhichbe the power of any assembly is increased. High local generated in the cladding offuelCANDU reactor fuel elements whenin vessel power ramping accompanies axial shuffling. High-burnup fuel rods type reactors canstartup also experience high-cladding stresses during rapid load changes or on after refueling. A number of fuel failures have been 2 This phenomenon is generally referred to as pellet attributed to this. 7 clfailaddin and fuel failures attributed to this are designated PCI uIrradiation res.g interaction of zirconium and its alloys increasesmusttheirbe considered hardness andin dede creases the ductility. This decrease in ductility a'
�w
PZr
T H E RMAL A N ALYSIS OF PWRS
1 20
signing reactortocomponents such as maximum pressure tubes. In fuelcladding elementstrain. design, it often leads a limitation on the allowable and mechanical properties of Zircaloy-4 are presented in Table 34 2.IVThermal A comprehensive property compilation is available in MATPRO. Sensitivity ofandthesuperheat 300-seriesconditions, stainless steels to stress corrosion cracking under boiling as well as the requirement for strength at high temperatures, has led to the consideration of high-nickel (up alloysto for72%supercritical reactors. Various Inconelforand300-series Incoloy compositions nickel as contrasted to 8% nickel stainless) and havehave been good proposed. These alloys arestrength. resistantAllioto stress-corrosion 2 7 cracking high-temperature and Thomas reported that Westinghouse 16-20behaved alloyssatisfactorily (16% chromium andfluxes 20% nickel with low carbon and nitrogen) at high inneuan environment of supercritical water. These alloys have a macroscopic tron absorption cross53 presents section ofthe0.design 266/ emproperties as compared to 0.300/ofemhighfor Incoloy. Reference of a number nickel There is noreactor longeruse.(1995) any significant interest in these ma terials alloys. for water-cooled High Nickel A lloys.
2 . 1 .2.2
Fuel Assembly Designs
Metallic, low-enrichment uranium fuel hasrods. almostTubing alwaysforbeen used indingtheisform of clad, solid or annular, cylindrical the clad commercially available, fabrication is relatively simple, and the nec essary surface areaas istheobtained bya proper selection of rod thesize.cladding When aluminum is used cladding, diffusion bond between and fuel iselements usually formed by a thin layerHanford of aluminum-silicon alloy.Reactor Zir caloy-clad were formed for the New Production (NPR)stripped by coextrusion ofextrusion. a copper-zirconium-uranium billet, with the cop per away after Rodareelements areformed also most commonly usedandforsintered uraniumU0dioxide fuel; they generally by loading pressed pellets into 2 Zircaloy or stainless steel* tubes to which end plugs are welded. Individual rods are2.6assembled intoof thebundles andassemblies end plates,for and fittings arewater affixed. Fisigngurewhere shows one earlier a pressurized de brazed ferrules hold the rods together. In more recent designs (Fig. 1.4),Thisthe rods haveallows been higher positioned by several grids containing spring fingers. design strength cold-worked cladding tofuelbe used, which reduces the amount of absorber in the core and allows the rods to expand, which reduces thermal stresses. Rods.
"All recent designs have used Zirca]oy, or other zirconium-based alloys, as cladding in order to minimize neutron absorption.
F U E L E L EM E N TS
---
-
1 1 1 . 25 i n .
1 00. 75 i n .
92 in. Fue l Assemb l y Center Line
--------
Fue l Tube Rubbing Strap Instrumentat i Pos i t ion - Control Rod
7.546 i n .
Fig. 2 .6
0 . 1 1 4 i n . Tube t o Tube
Original Yankee fuel assembly [from Directory of Nuclear Reactors, Vol. IV, p. 34, International Atomic Energy Agency, Vienna (1 962)].
Asforpreviously noted, the newer assemblya designs also eliminate the cut outs cruciform control rods. Instead, typical 1 5 X 1 5 rod assembly would contain 20 hollow tubes serving as guides for rod cluster control (RCC) elements. Eachloaded fuel rodintocontains a stack of pelshortlets(e.areg., centerless 0.7-in.-Iong)ground pellets.to Prior to being the claddi n g, the an out side diameter, which is about 0.007 areto 0.usually 008 in. lessground than theto provide claddingdished inside diameter. In addition, the pellets ends. During reactor operation the center region ofisthedesigned pellet istohotter and expands more than the outer region. The dishing produce nearly flatgaspelletvolume ends(plenum) at power.is provided in the upper few inches of each A free fuel rod.operation The plenum volume is excessive sized so that the pressure fission gases released during do not produce internal at the endof theof life. A hold-down spring pl a ced in the plenum ensures that handl i ng fuelTherodrodswillarenotgenerally result in initially movementpressurized of the pelletwithstack. helium. This improves the heat transfer across the pel l et-to-cladding gap and reduces cladding creep-down due to the external pressure.
TH E RMAL ANALYS I S OF PWRS
Solid pellets have(1995). beenHowever, used in hollow the fuelannular rods ofpellets the vasthavemabeenjorityusedof operating reactors in Vibratory the Russiancompacted VVER reactor fuel elements. U0 has beenofconsidered as a substitute forcom sin 2 tered pellets. Because of the difficulty producing very-high-density pacts and thecladding greaterpinholes, tendencyvibratory of vibratory compacted material to wash out through compacted fuel has not been used (1994) for operating plants. The need to fabricate PU0 -U0 fuel elements 2 2 remotely has,V provides however, typical led to renewed interest in thisrodapproach. Table 2. assembly and fuel dimensions forlatera 1517 1517 assembly designed for a l a rge (800to 1200-MWe) PWR. In the assembly designs, rod pitch and rodin Table and pellet diameters are re duced by about 10% from those indicated 2. V Fuel assemblies designed for used the CANDU pressurePWRs. tube reactor are conan siderably different from those in vessel-type Since such assembly mustfuelfit within a pressure tube, ithave contains onlyshort a smallwithnumber19lengths to 37-of rods. These assemblies all been bundle no more than six times the diameter of the rod cluster. In one deof sign, a completely integral structure with end plates welded to the ends 1 Spacing between the fuel rods is maintained by 7 allmetalthewarts pins isbrazed provided. obliquely toferrules, the fuelandpins.spacer In other designs for pressure tube assemblies, wire wraps, pads have been used tois maintain rod spacing. A drawing of a typical CANDU fuel assembly shown in Fig. 2.7 Designsdispersion for highlyelements enrichedareuranium-Zircaloy ele ments and U0 -stainless-steel similar. In both cases, 2 the so-called "picture frame" method is used for fabrication, which means X X
Plate- Type Elements.
TABLE 2.V Typical Assembly and Fuel Rod Di mensions for Large PWRs
Rod array Fuel rods per assembly (20 RCC thimbles, 1 in-core flux detector thimble) Assembly length Rod pitch Rod o.d. Pellet o.d. (cold) Cladding thickness Cladding material Overall fuel rod length Active fuel length Number of support grids Length of gas plenum
15 X 1 5 204 1 59.7 in. 0.563 in. 0.422 in. 0.366 in. 0.024 in. Zircaloy-4 151 .8 in. 1 44 in. 7 (6 in active core) �5 in.
F U E L E L E M E NT S
F i g . 2.7
1 23
CA:'>J DU fuel element: (1 ) Zircaloy structural end plate, (2) Zircaloy and cap, (3) Zircaloy bearing pads, (4) uranium dioxide pellets, (5) Zircaloy fucl shea th, (6) Zircaloy spacers [from ]. A. L Robertson, AECL-4520, Atomic Energy of Canada Ltd . (1 973)] .
that a Upper sheet ofandthelower fuel alloy, orofdispersion, is rimmed by a frame offrame clad ding. sheets cladding are then welded to the and theinpacka box-like is rolledenclosure to final dimensions. A groupmaterial. of fuel This platesmaintains are then placed made of cladding plate spacing and provides an assemblyfuelthat canin Shippingport be handled. Plate-type elements containing uranium-Zircaloy used Core Idisare -stainless-steel illustrated in Fig. Plate-type elements using a U0 2 persion as fuelelements have beenforused infuelthe have Armybeen Package Power Reactor (APPR). used in Shippingport Core Plate-type U0 2 II. Rectangular platelets of U02 fuel are placed in a slotted Zircaloy plate, which isZircaloy sandwiched between two Zircaloy cover plates.under The fairly sub stantial plates served to restrain the U0 platelets high burn 2 up. Suchtheplates can be fabricated by the gas-pressure bonding process5 in which fuel element can be assembled from plates and strips that are subsequently bonded together at high arepressures and considered temperatures.for use in Although plate-type fuel elements no longer central powerfuelstation cores,cantheyprovide are obviously ofpower interestdensity for naval reactors. Plate-type elements the high and compact cores needed in this application. 2.8.
T H E RMAL A N A L Y S I S OF PWRS
1 24
Boron- Containing Plates, 24 p er Element Standard Plates 44 per Element c -0 -0 0
a
C
"
Control Rod passage
11
-0 M
� 0
• c -0 -0 0
a
Fig. 2.8
C -0 M 0
a
2.0 2.2
in. in
• .
5.6
in.
Shippingport Core 1 seed assembly [from Directory of Nuclear Reactors, Vol. IV, p. 21, Interna tional Atomic Energy Agency, Vienna (1962)].
Tubular fuel elements can bestainless-steel fabricated fromformi sandng wiches of dispersion fuel and cladding. Standard 4 Elements operations can bein used to makereactors the sandwich sheets? of US. this tArmy ype were used a portable (PM series) designed for the nuclearfuelpower program. Annular elements, such2 pellets. as thoseVarious of the firstvibratory USSR power station,tech can compaction beniques formedcan from hollow U0 also annular be used elements for annularin individual fuel elementpressure fabrication. Multipass, concentric ring, tubes have been proposed for supercritical pressure reactors?5 The newer assembly designs forTheseconventional PWRsbeenincorporate a order numberto ofreduce significant improvements. changes have made in fuelandfailure frequency, allow extended burnup, increase thermal margins, improve neutron economy. A typical advanced fuel assembly is shown in Fig.advanced designs have improved bottom nozzles, which are All of the designed to block debris entry and thus byreduce debris-caused failures.in theIn many cases, this is simply accomplished using small flow holes nozzle instead of theislarge size openings previouslyanused. In someantidebris designs, the debris removal enhanced by incorporating additional filter in the form of a screen under the nozzle. Alternatively, the lower Tubular Fuel Elements.
2 . 1 .2.3
Advanced Fuel Assembly Designs Using Cylindrical Fuel Rods
2.9.
F U E L E L E M E NTS
Fig.
2.9
Typical advanced fuel assembly design (courtesy Westinghouse Electric Corp.).
1 26
T H E RMAL A N A L Y S I S OF PWRS
nozzle platesomayas toconsist of a debris parallelentry. array of closely spaced curved blades configured preclude In been most replaced advancedbyassembly designs, theIn stainless or Inconel spacer grids have all-Zircaloy grids. one design, the grids have been replaced by Zircaloy but the springs holding the rods in place are Inconel. There is some evidence thatforZircaloy springsavoids tendthisto relax duringAdditional irradia tion and the use of Inconel the springs problem. absorber is removed from thechanges assembly by usingreduce Zircaloytheforparasi the tcontrol element guide tubes. These significantly ic neu tronIn absorption and allow lower fuel enrichments to be used. essentially allvanes of thethatadvanced assembly designs,mixing. the spacer grids con tain small mixing enhance interchannel Most advanced designs also portion contain ofintermediate flowThese mixerlow-pressure grids in thedrop high-flux region ofprovide the upper the assembly. grids do not rodthesupport butwheredo contain mixing vanes thatof increase interchannel mixing in region the critical heat flux is concern. They there foreA provide additional thermal margin. removable uppercontains nozzle isfailed used fuel in allelements of the advanced assembly de signs. If an assembly and has a fixed upper nozzle, the entire assembly must betheremoved duringforrefueling. Ultrasonic testing techniques, which compare time required a signal to proceed around the cladding of failed and unfailed elements, now allow individual failed rods toonlybe those locatedrodswithinthatanhaveassembly. Therefore, with a removable top nozzle, failed need to be removed. The as sembl y may then be reconstructed, using replacement rods onl y for the failedThefueltop elements, and then returned toflexible the core.clamps and/ or spring as nozzle is now provided with sembliesIn athatfewholdassembly the assemblies firmly inareposition against thethe upper core plate. designs, springs used to hold individual fuelTherodsfuelin rods position against the thetop assembly nozzle. have also been modified to that constitute allow forassembly higher burnup and greaterusedneutron economy. The early spring finger designs generally a 15 15 array. In response to LOeA considerations, theto about array 0.375 was subsequently increased todesigns 17 17 and rod sizes were reduced in. More recently, some have reduced rod sizes to about 0.36 in. The increased moderator-to-uranium ratiocosts that isthisachieved producesby slightly reduces fuelleakage costs.through A furtherthereduction in fuel reducing neutron use of axial blankets of natural orcontrol depletedof reactivity uranium oxide pellets.of that provided by the In earlier designs, in excess control rods wasguide provided byExcess burnable absorber rodsrequirements placed in unused control element tubes. reactivity control areIn inall creased in the advanced designs because of the high burnup desired. of the advanced assembly designs, excess reactivity control is provided by X
X
1 27
F U E L E L E M E NT S
burnable poisons that coat or are mixed with the V02 of some fuel rods (seeHigh Chapter I, Sec. 1 .2.3.3). fuel burnup leadsrods.to increased fissionfission gas release and isincreased axial growth of the fuel The increased gas release accom modated by increasing the length of the gas plenum. Increased rod growth istheaccommodated by increasing the clearance between the top nozzle and rods. The assemblies describedPWRs.in thisAs indicated section havein Chapter been designed for use in conventional vessel-type 1 and Sec. fuel for pressure-tube and offorthistight-lattice, coresassemblies differ substantially from thecores designs section. high-conversion 2. 1 .2.2,
2.2
B E HAVIOR OF U 02 F U E L ELEME NTS
Sinceusedfuelinelements consisting of V02thepellets in cylindrical Zircaloyof tubes are nearly al l power reactors, special design properties these elements are examined.* 2.2.1
Mechan ical Properties of U02
Atto ductile temperatures belowtemperature, 1000°C, V02 isfracture a brittlestress material. Below the brittle transition is smal l e r than yield stress and brittle fracture occurs. Hence, at relatively modest thermal cracking V02 pelletE, occurs. the stress-strain curve isstresses, linear and Young'of sa modulus, can be Below obtained from76 E2 X 104 X 04 where E isE independs N I mm , is in K, and porosity is given as a fraction. The value of slightlyfromon the oxygen-uranium ratio. If thein Eoxygen uranium ratio is increased 2. 0 to 2.1 6, an decrease can be 6 7 expected. Siinsensitive nce the fueltois themuchvalue stifferofthan the cladding, fuel Some designcomputational computations are E used for the fuel. 2 procedures, therefore, take E as a constant (e. g . , X 0 N 1m ). Thedatatypical value ofonPoisson' s ratio, dependence, is reportedbutto other be 0.31ceramic 6 (Ref. 76). Noterials are available temperature ma show little effect of temperature on The dependence on porosity is very small. �
To
To
= 22.9
- 20. 1 T - 58.7
T
1
a
(2.24)
a
�22%
2
1 11
"V ,
'Y.
*Interest in burning plutonium removed from nuclear warheads could lead t o some use o f ceramic fuels t h a t are not primarily V02 • T o avoid production o f additional P u during irra diation, the Pu would have to be dispersed in an inert material. Akie ct at. [NIlc/. Techno!. 107, 1 82 (1994) ] indicate that a mixture of Pu02-AI201 and stabilized zirconia (Zr02 plus small amounts of Y02 ) should be suitable for once-through burning of weapons plutonium.
1 28
T H E RMAL ANALY S I S OF PWRS
of isdepends Fracture stressof porosity onandporosity, grain size,Both and tempera tempera ture. The effect smal f 6 can be neglected. ture and grainassizeto have significantofeffects, althoughStehle thereetseems to be some 7 disagreement the magnitude these effects. a1. 6 suggest that fracture stress can be expressed by (2.25) 770.0and where is inthatN /mm2varies and fromis in�98.K.5Evans andandOavidge Canon al?The 8 indicate to 151 from 25 to 0.0equal 28. brittl e to ductile transition temperature, is approximatel y to aboutporosity, half thegrainabsolute mel tingstraintemperature. The orvallow ue ofstrain Te also varies with size, and rate. At zero rates, varies from � 1000 to 1100DC. Increasing the porosity of the pellet slightly decreases A second transition temperature, istensile definedstressas measured the temperature that shows a marked decrease in the ultimate in bending tests. In the(semibrittle intermediate range,For< T < thesomefracture deformation occurs before fracture fracture). mode is compl etely ductil e . The value of depends on strain rate and porosity. At low strain rates, is 1800DC on the (Ref. order78).of 1400DC, while at very high strain rates it can increase to and simplthater belwayowofarepresenting the brittle-plastic behav iorAnofthealternative is to assume temperature-dependent yiel d stress, material is elastic and stress and strain are proportional. Above � the fuel is taken as perfectly plastic. Krammen and Freeburn48 correlate � by forfor where yield strength Pa fuel temperature, DC plasticity onset temperature1 0 = 1200DC D = yield coefficient 1. 6 10 , Pa DC. At high oxide temperatures, exhibits significant thermal creep. and However, fissioning fuels exhibit significantly enhanced pl a sticity show fission-induced creep at temperatures where thermal creep does not occur. 7 Olsen 9 presented anandempirical correlationor enhanced of the datacreep. on thermal creep and combined thermal fission-induced This correlation, as revised by Hagrman et a1.34, is given by ITf
U02
IT f = (J f() + AT
ITf
T
IT,
A
III
TC f
Te
Te .
Tt ,
Te
Tt ,
T > Tt ,
Tt
Tt
U02
ITl/ ,
(J , IT I
T ::s Tpa
T > TPa
IT�t '
et
U02 ,
T =
TPa =
=
U02
X
1 29
F U E L E LE M E N TS
(2.26)
where 3 AA7 == = creep rate, m/m· s 0: = compressive stress, Pa 1= F = fission rate, fissions I (m3 s) A 2 AA3 == QG3 == grain size, fJ.J/mol m 46 = constant, J I mol K A RT == gas temperature, K mol ratio d = density (fraction of theoretical). x = oxygen-to-metal Also Q1 { exp [ log(x Q2 = { exp [ log(x When stressed, plastic flowpressing. of V02 "canStresses lead toondensification of arise fuel pelwhen lets through "hot hydrostatic the V0 can 2 fuel swellingSuchbrings the outer diameter ofpeltheletfueljamming pellet inatcontact with the cladding. stresses can also arise if some elevation prevents freeby axial expansion ofhotthe pressing stack. Iniscontrast to creep, which is governed deviatoric stress, governed by hydrostatic pressure. Althoughpressing), severalitmechanisms contribute toumepressure sintering (hot dif hy drostatic woul d appear that vol creep due to lattice fusion mechanism for V0et2a1.and34 asMOX fuel. The available data haveis thebeendominant represented by Hagrman � ddtpl = A ( P Pl )I1--; P1 1 Tg exp[Q/(RT)] , where a = for D02 , for MOX fuel t time size, fJ.m g == grain fraction of theoretical density P1 3.72 X 1 0 - 5 0.3919 1 .3 1 X 1 0 - 87.7 2.039 X 1 0 - 90.5
£,
2.61 7 x 1 03,
= 1 7,885
1 9,872
- 20
2)
- 20
_
1-
=
2.7
2.55
2)
+ 72,1 24
+ 1 1 1 ,544
(2.27a)
T H E RMAL ANALYS I S OF PWRS
1 30
P = = T = =
A R For
hydrostatic pressure, Pa
U02 ; K J I mol K.
48940 for
1 .8 X 1 07 for MOX fuel
temperature, 8.314,
U02 ,
[
the pressure sintering activation energy,
Q=
R{
9000 exp
20 - 8110g(x - 1 .999)1 Ilog(x - 1 . 999)1
+ 1 .0
Q,
]
-1
is given by
+ 36'294
}
(2.27b)
where x is the oxygen-to-metal ratio. For MOX fuel, Q = - 45,000. Since reactor fuels are close to the theoretical density of U02 , the left hand side of Eq. (2.27a) equals the volumetric strain rate (dChp I dt ) . Further, the volumetric strain is readily reduced to linear strain by noting that the hot pressing strain components are all equal
[(Chp )6 = (Chp )r = (Chp )z ] = (1/3)(Chp ) 2 .2 .2
Fuel Densification and Restructu ring
Hot pressing is not the only mechanism that can lead to in-pile densification of U02 . Pellet densification and shrinkage can be caused by irradiation induced dissolution of pellet porosity. Many investigators believe densifi cation results from dispersing fine pores as vacancies by fission fragments. These vacancies subsequently diffuse to a grain boundary leaving a den sified fuel grain. 80,81 The process is illustrated in Fig. 2.10. Irradiation-induced fuel densification is of greatest importance in low density (e.g., � 92% theoretical) fuels. When low-density fuels were first used, irradiation-induced densification caused pellet shrinkage that led to gaps in the pellet stack and cladding collapse in the space between pellets. The problem is now understood; reactor vendors control densification by limiting the initial volume of fine pores in the fuel.81 Increased sintering temperature and sintering time lead to a significant decrease in densifica tion. Several approaches have been suggested for estimating the rate of den 2 sification. Rolstad et al.8 suggested a relationship between percent densi fication, sintering temperature (a measure of initial pore and grain size), initial density, and burnup. The graphs required for the use of this model are reproduced in Ref. 83. Marlowe84 explains irradiation-induced densification by an irradiation enhanced diffusion model. It follows Coble' S85 theory for sintering, which is based on the assumption that the driving force for pore shrinkage is the effort to reduce free energy by lowering pore surface area. According to Marlowe, in-pile pellet density, p, is obtained from
P = Po
.) . M [ - S ( _G0_3. + Ft. ) ] ( + AD. -----jf Ft A ADm Go
exp( - SFt ) + - exp
In
1
,
(2.28)
F U E L E L E M E NT S
Porous U 02 G ra i n { Many F i n e Pores Repre se nted b y One Circu lar Pore at the Center of a G ra i n }
J
H omog e n izati on by Fi ssi on Fragments (Arrows) D i spe rse the Pore as Va canci es (. Marks)
D i ffusion of Va canci e s from Poi nts of H i g h Conce ntration to G ra i n Boundaries.
j
Densifi cation of Or i g i na l l y Porous G ra i n by E l i m i nation of Va ca n c i es at Externa l G ra i n Surfaces.
Fig.
2 . 10
Schematic illustration of in-reactor pore removal [from Ref. 81].
where
Po
=
�= F=
M=
Din
=
initial density grain growth constant fission rate densification rate constant constant relating irradiation-induced diffusivity to fission rate
TH E RMAL A N A L Y S I S OF PWR S
5
= volumetric swelling rate for 1 00 % dense fuel
t=
Go =
irradiation time initial grain size.
M A
0 i rr 0irr 10'
Quantities and can be determined from experiments on the thermal sintering of the pellets under study. Buescher and Horn86 suggest that can also be calculated from thermal sintering data by using = 1 . 1 5 X 1 0 .- 1 5 cm3 s / fission, where i s the thermally activated bulk dif fusion rate. Stehle�l7 points out that the net densification observed is the algebraic sum of fuel swelling and densification. Further, experimental studies show that the fine pores shrink very much faster than the larger pores. He con cludes that the net densification can be expressed by a semiempirical equation
0'
(2 .29) where � V I Va Bu ( PO ) i ( i
ro )
(x'
= = = = =
fractional change in volume burnup (GWd / tonne) initial pore volume in pore class average radius of pores in class constant.
i
i
The foregoing equation leads to a more rapid disappearance of the finer pores. A pore class disappears when the bracket in the equation becomes unity. In view of the improved behavior of current fuels, some more recent approaches tend to use simpler, more empirical approaches. In-pile data are used to establish ( 1 ) the total amount of densification possible with a given fuel and (2) the rate at which this densification occurs. This latter quantity can be expressed as a rate constant47 or in terms of a burnup at which the densification is complete.48 With the latter approach we have: �V
11 = (�p)o { exp [ Bu o
In( O.OI ) / (Buo ) ]
-I}
(2.30)
where (�p)o is the total densification that can occur as a percent change in fuel density and BUD is the burnup at which densification is complete. Den sification occurs fairly quickly and BUD is about 5000 MWd / tonne for 750°C. The value o f BUD appears to increase a t low temperatures. Thermal effects also lead to in-pile restructuring of U02 . In Sec. 2. 1 . 1 , we noted that porous oxide fuel, irradiated a t high heat fluxes, shows large columnar grains distributed around a cylindrical central void. In out-of-pile
T�
F U E L E L EM E NTS
experiments, MacEwan and Lawson88 showed that columnar grains would be produced at temperatures above 1 700°C if a sufficiently high temperature gradient were maintained. Columnar grains resulted from the migration of voids, transversely oriented with respect to newly formed grains, toward high-temperature regions. These grains therefore have their long axes parallel to the radial temperature gradient. The observed grain growth is attributed to a vaporization-condensation process that results in a net movement of the pore at high temperatures. We assume that fuel vaporization occurs at the high-temperature end of the pore and condensation occurs at the low-temperature end. Nichols89 developed a theoretical model for this process. More recently, considerably simpler correlations for the rate of pore movement have been proposed. Hagrman a1. 34 correlate the pore movement by
et
u
49.22(V T)
exp( - 44980 I T )
(2.31 )
= ------=--------
T2
where u
VT T
= rate of pore movement, m/s = temperature gradient, Kim = temperature, K.
Nichols has taken the region in which all pores have had sufficient time to migrate to the central void as the columnar grain growth region. By using Nichols' model or Eq. (2.31 ), we can compute the radial boundary of the grain growth region, reg . Figure 2. 1 1 presents the results of Nichols' calcu lations for a typical U02 pellet. He assumed the surface temperature of the pellet was 800 K and that a parabolic temperature gradient existed across the fuel. Helium was assumed to be the primary vapor constituent. Observe that reg can become a significant fraction of ro , the total pellet radius, at high temperatures. The radius of the central void, rV I is computed by (2.32)
The columnar grain region is surrounded by an outer region of equiaxial grains. These are enlarged fuel grains having all sides of approximately the same length. The rate of equiaxial grain growth during any time interval can be calculated from34 G'
=
[
1 .0269 X 1 0 1 3 t
]
1f4 exp( - 35873 I T ) + G04 0 .0 - 5.746 X 1 0 - 6 BU)2T -
where G ' = grain size at end of time interval, m G o = grain size at beginning of time interval, m
(2.33)
1 34
T H E RMAL A NA L Y S I S OF PWRS
Note: To Convert to Radi u s of Resu lti ng Centra l Vo id, Multi pl y rcglrO By "Porosity Fraction
3500
Un i rradi oted U 0 2 Me l t i ng Po int
5840
4940 .. �
u.. 0 :::>
4040
eQ)
Q.
E
Q) �
3 1 40
1 500 .
o
Fig. T
Bu
t
= = =
�_�_�_.L..-_....i-_ ....i-_...L. _-'-----..--.J
__
0. 1
2.1 1
0.2
0.3
0.4
0 .5
0.6
0.7
2240
0.8
Columnar grain growth radius in U02 rods [from Ref. 28] .
temperature, K burnup, MW·s/kg time interval, s.
The outermost region of the pellet is as fabricated unstructured fuel. This region will, however, show significant thermal-stress cracking. At the relatively low linear power at which current PWRs operate, no columnar grains or central void are formed. Central temperatures at steady state operation are now (1995) generally no higher than 1 800°C. Columnar grains may grow during short-term transients but they do not strongly influence rod behavior. Hot-cell examinations of irradiated PWR fuel sub jected only to normal operation show only an outer unrestructured (but cracked) region and moderate growth of equiaxial grains in the center of
F U E L E L E M E NTS
the rod. Since only equiaxial grains are normally seen, Eq. (2.33) may be used to compute the grain size needed for densification and hot pressing estimates. At low fuel temperature, Eq. (2.33) will simply indicate a grain size close to the fabricated size. If the rod has been subjected to a high power transient, a region of shattered or desintered grains, consisting of fuel grains that are fractured free of bonds to other grains, may be seen. 2 . 2 .3
Fuel Swel l ing
Previously we observed that uranium dioxide swells during irradiation. The data indicate that the rate of fuel swelling is a function of temperature. The swelling initially fills a portion of the fuel porosity. When the portion of the porosity that is effective in accommodating swelling has been con sumed, net dimensional changes occur. Table 2.VI indicates approximate swelling rates and effective porosities for several temperature ranges.90 The burnup at which the effective porosity is consumed is similar in concept to the critical burnup for rapid swelling (see Fig. 2.1 ). Godesar al.9 1 suggest an analytical expression for total swelling, which was calibrated to a series of in-reactor experiments. They propose
et
� V I Va = fa
where
[ ( F �:) ] exp -
(2.34)
- 1 + [3(1 - al b
= b = a = B = =
� V / Vo
relative volume expansion by swelling burnup, MWd / l000 kg U initial porosity swelling rate of 1 00% dense oxide 1 .6 X 10 - 6 (MWd/ l 000 kg U) - l for usual PWR fuel temperatures f = density correction factor = 0.149(a) - 0 . 746 F = geometrical factor = 1 00.
This model takes into account the influence of porosity on swelling rate. As the volume of pores decreases with time, due to inner swelling, the TAB L E 2.VI Approximate U 0 2 Swell ing Rates and Effective Porosities
Temperature Range
Fuel Swelling % (dV / Vo) per 1 04 MWd / tonne
Effective Component of Initial Voidage (%)
< 1300°C 1 300 to 1 700°C > 1 700°C
1 .6 1 .7 0.7
30 50 80
1 36
T H E RMAL A NALYS I S OF PWRS
apparent outer swelling rate increases toward an asymptotic value. Al though Eq. (2.34) has generally been supplanted by more complex models, it does provide a simple expression that is moderately accurate and de scribes the major trends observed experimentally at modest burnup. Recent approaches to fuel swelling have been more mechanistic. Fuel swelling is generally attributed to production of bubbles by gaseous fission products and the accumulation of solid fission products. Solid fission prod ucts produce swelling of between 0.5 and 1 % per at. % burnupY 2,93 The contribution of solid fission products to swelling is most important at low ternpera tures. At high temperatures, gaseous fission products are the major cause of fuel swelling. By calculating the volume of gas bubbles within the fuel, the increase in fuel volume and, hence, swelling due to fission gas, can be estimated. Several models have been proposed for this purpose. The first fairly comprehensive model was Nichols and Warner' s BUBL-l code,94,95 which calculates the nucleation of gas bubbles, their coalescence, interaction with grain boundaries and dislocations, and migration of bubbles due to the fuel temperature gradient. The model assumes the following: 1 . Bubble sizes are determined by gas pressure balancing surface tension and hydrostatic pressure. 2. There is a critical force required to pull a bubble away from a dislo cation or grain boundary. 3. Bubble migration along the temperature gradient is due to a surface diffusion mechanism. Fission gas behavior is followed by a Monte Carlo technique. When two bubbles overlap over a dislocation, they are assumed to coalesce. When a bubble grows sufficiently, the driving force due to the temperature gradient pulls it off the dislocation and the bubble migrates to a grain boundary by surface diffusion. The bubble remains at the grain boundary until further coalescence allows it to grow large enough to be pulled away. The bubble then migrates along the temperature gradient. Release of the gas is assumed to occur after the bubble has gone a distance approximating the distance to a crack or surface. Yu-Li al.96 and Rest97,98 describe a somewhat similar model. However, their approach, which is embodied in the GRASS code, is somewhat more general. The general approach of GRASS is basically the same as BUBL in that coalescence, release from defects and boundaries, and migration along the temperature gradient are followed. However, the model allows for ra diation-induced resolution of the bubbles as a negative term in bubble growth. The GRASS code also allows for random motion of bubbles and coalescence during migration. Surface diffusion is not the sole method for bubble diffusion. Diffusion components are calculated on the basis of sur face diffusion, volume diffusion, and evaporation-condensation. In each
et
F U E L E L EM E NTS
fuel region and for each bubble size, the mechanism giving the highest bubble velocity is chosen. Recent models of GRASS also account from the fact that the capacity for storage of gas bubbles at the grain boundary is limited. The GRASS progrc:;n is slow running since the code uses a bubble size distribution, which changes through bubble coalescence and resolution. These changes are determined by solution of a large number of coupled nonlinear differential equations. To avoid this problem, the FASTGRASS program98 uses an average bubble size and characterizes intergranular, grain face and grain edge bubbles by number density and an average num ber of atoms per bubble. Gas release from the grain boundaries occurs only when a specified fraction of the grain boundary area is covered by bubbles. This leads to running times that are only a fraction of those of the original GRASS program. The GRASS series of codes is considered to be the (1993) state-of-the-art programming for fission gas swelling and release from LWR fuel by the U.s. Nuclear Regulatory Commission. The detailed models of fission gas behavior determine the fission gas released, the fission gas dissolved in the crystal lattice, and the fission gas contained within the bubbles located at dislocation and grain boundaries. Fuel swelling is determined by the latter quantity. By assuming that the trapped gases are located primarily on the grain boundary, MacDonald al.99 computed the fission-gas-induced swelling (a v / V)g , due to grain boundary gas accumulation, as:
et
(2.35) where Nb = grain face bubble density, m X =
R =
r =
ex =
e =
fuel surface area to grain boundary area average grain radius, m grain face bubble radius, m [1 (3 cose - cos3e) / 2] sin3e semidihedral angle of lenticular bubbles =1T /3. -
Despite the improvements in running time offered by FASTGRASS and other recent codes, such as OGRES,1 00 many analysts consider such models to be too time consuming for routine design use. Considerably simpler swelling models are therefore often employed despite the present availa bility of well-benchmarked mechanistic approaches. The ESCORE code48 is typical of many current design programs. ESCORE computes the swelling as a sum of solid and gaseous swelling less a portion accommodated by the fuel porosity. The solid-fission product
1 38
T H E RMAL A N A L Y S I S OF PWRS
swelling is taken as directly proportional to burnup and independent of temperature. (2.36a) where a is the porosity fraction and Bu is the burnup (MWd / tonne). The fractional volume change due to gaseous fission product swelling is taken as: (�V I V )g = G (1 - a)(�Bu)
(2.36b)
where G is the swelling rate for fuel in specified axial location for given time step and for 650°C ?: Tavg > 21 oooe for 650°C � Tavg < 1 1 oooe for 1 1 000e � Tavg � 21000e The average fuel temperature, Ta vg , is for a given axial segment (CO). Note that the foregoing gas-induced swelling approach is very similar to that of Table 2.VI. In using the previous two equations, it is assumed that a portion of the swelling computed may be accommodated within the fuel pores, fuel pellet dishes, and radial pellet cracks. The volume so accommodated, (�V I V ) A ' corresponds t o 1 0 % o f the solid fission swelling and 50% o f the gas induced swelling. That is, (2.36c) where (�V I V)N is the net or observable swelling. When the absolute value of (�V I V )N equals the total volume available for swelling accommodation, no further accommodation occurs. The use of pellet dishes and fuel crack volumes for swelling accommodation differs from the usual approach, which accommodates swelling only in fuel pores. The ESeORE model of swelling is conservative in that the fission-gas induced swelling is always directly proportional to the burnup. Many of the current mechanistic fission gas release and swelling models postulate that, after the concentration of fission gas bubbles on the given boundaries reaches a limiting value, the grain boundaries are all interlinked. Any fis sion gas subsequently arriving at a grain boundary is therefore released. The bubble concentration at the grain boundary would then reach a limit at high burnup. Since gas-induced swelling depends on this concentration, the swelling would reach a maximum value. This idea is embodied in the swelling computations of eOMETHE 1114 7 and SPEAR. 1 O I The relationship used to determine the increase in pellet porosity due to gas bubbles is: (2.37)
F U E L E L E M E NTS
where
a
(Bu) L Bu aF aL
= = = = =
1 39
(aL - aF ) / (l - a L ) burnup constant current burnup as fabricated porosity (pellet void fraction) limiting porosity (function of applied hydrostatic stress).
Equation 2.37 predicts that the total gas-induced swelling will asymp totically approach a. This is in contrast with Eq. (2.34), used in earlier COMETHE versions, which indicates an asymptotic "swelling rate." The foregoing model computes the externally observed swelling since the fab ricated porosity is subtracted from the limiting porosity. An approximate value of the limiting fuel matrix swelling in the absence of applied stress may be obtained from Eq. (2.35) by making use of values for bubble size, r, and grain bubble density when interlinkage of grain boundary bubble occurs. MacDonald a1.99 suggests that under these conditions
et
Nb=2 X 1 0 - 7 m2 and r = 2 X 1 0 - 7 m The limiting externally observed swelling would then be determined from the computed (d V / V )g by making an appropriate allowance for swelling accommodation within the fuel pores. The idea that grain boundary bubble interlinkage leads to a maximum swelling value would also appear to apply to metallic fuels. In-pile exper imental observations of metallic Pu-U fuel elements designed for fast re actor use show that swelling rates are initially higher than for oxide fuels. However, a maximum swelling is reached and further increases of burnup do not cause a further increase in fuel dimensions. By designing the fuel rod to accommodate the maximum swelling, metallic fuel elements capable of withstanding very high burnups can be fabricated. Net swelling is seen as the algebraic sum of irradiation-induced densi fication and fission product swelling (2.38) Therefore, a small amount of fuel densification may not be injurious since it may partly counteract the effects of swelling. 2 . 2 .4 2.2 .4.1
Fission Gas Release and I nternal Pressu re Fission Gas Release Models for Normal Operation
Gaseous fission products are generated at a rate of approximately 0.3 at oms / fission. A significant fraction of the gases so produced is released by 002 . It is generally agreed that below �600 to 800°C, the rate of fission gas
T H E RMAL A N A L Y S I S OF PWR S
1 40
release is controlled by recoil and knockout.102, 103 Release rates are low and not temperature dependent. In the region between 800 and 1 800°C, fission gas release rates are higher and markedly temperature dependent.* For a number of years, the rate of gas release in this region was believed to be determined by the rate at which gas diffused out through solid U0 particles. Available data were correlated 2 using a fission gas diffusivity, 0, which followed an Arrhenius rate equation 0 = 00 exp(
-EIRT)
(2.39)
where 00 = = = =
E R T
a constant activation energy, cal! g mole perfect gas law constant, 1 .986 cal l g mol, K absolute temperature, K.
According to Booth,1 04 activation energy is essentially constant, but 0 0 varies with the nature of the compact. By using this theory, he developed a method for computing fission gas release based on the diffusion equation for a spherical particle of radius a. He expressed his results in terms of F, the fraction of total gas produced that has diffused out of the sphere in time t. From t = O to t = 1 / (1T2 0' ) (at which time F = 0.57), fractional release can be approximated by F=4
(� r ( � ) t
_
3 't
(2.40)
where 0' = 0 I a 2 To account for later data showing that fission gas release is burnup dependent, some subsequent diffusion models have multiplied the right-hand side of Eq. (2.39) by a factor that depends on burnup and grain size. In most instances, the designer must consider conditions in which the fuel temperature varies with time and can be considered constant for only short periods. For a short period of constant temperature, the gas release may be determined by considering the change in the average concentration of the fission gas in the particle. Verbeek and Hoppe47 approximate the solution for the average gas concentration in the Booth particle at the end of time step (at) as:
*Bagger, Mogensen, and Walker [J. Nucl. Mater 211, 11 (1994)] found that, at actual reactor operating conditions, fission gas release by U02 pellets is negligible under about 1 200°C.
1 41
F U E L E L EM E N TS
(2.41 ) where
R (A) I 1T4) L e An2) I n4 = (90
11 = 1
(1
-
-
and C t is the average gas concentration in particles (mol! cm3 ). The fission gas release, G, during the time step is then (2.42) where � is the fission gas production rate (mol! cm3 s) and G is the fission gas release rate per unit volume (mol l cm3 fuel s). Later studies indicated that the fission gas release process is much more complicated than originally believed. It has been suggested that a coupled diffusion-trapping process is the rate controlling mechanism.62 In such a model, fission gas is assumed to be trapped in lattice defects; its release is partially controlled by its rate of escape from these traps. More recent mod els have been of the bubble migration type, which have been used for pre dicting fuel swelling. Thus, the previously described BUBL (Ref. 94) and GRASS (Ref. 97) codes can be used for predicting both fission gas release and swelling. However, a critical survey by Ronchi and Matzke1 05 estimates that most of the fission gas is kept in the matrix by resolution and that atomic diffusion accounts for a large fraction of the release. This view is now generally agreed on, so models like BUBL-1, which describe fission gas release solely in terms of bubble migration, are considered inappropri ate. Models like GRASS, which account for trapping resolution and atomic diffusion, do appropriately describe the release process when calibrated against experimental data. Although the GRASS codes were initially developed to provide a de tailed phenomenological model for fast reactor fuel, they have been incor porated into such fuel design codes as the LIFE code for fast reactor fuel and FRAP for LWR fuel design. While the GRASS codes are considered ( 1 993) by the U.s. NRC as pro viding the most satisfactory "best-estimate" fission gas release model, 106 the OGRES steady-state computer model appears to have an equivalent position in Great Britain. The OGRES-I model determines the gas arrival at the grain boundaries as primarily due to single gas atom diffusion. The trapping of gas atoms by intragranular gas bubble and resolution of the gas by fission fragment destruction of the bubbles are in essential equilib rium. However, migration of large bubbles to the grain boundaries is con sidered. Release of the gas at the grain boundaries occurs primarily by gas
1 42
T H E RMAL A N A L Y S I S O F PWRS
P ro d u c t i o n 01 -
-
j
I
O s s i o n g as ---
B u bble nuc leat i o n a n d g row t h D i ss o l ve d
Gas b u bbles
g as
inside grains Re s o l u t i o n 0 1 g a s by
V
f i s s i o n 1 r ag m e n t s Gas d i Uusion B u b b l e m i g r at i o n to g r a i n b o u n d a r ies
t o g ra i n b o u ndar ies
Res o l ution of g a s f ro m g r a i n b o u n d a r i es
G rain bou ndary b u b b l es
R e l e as e of g as qt g r a i n b o u n d a r y
Fig.
2.1 2
Fission gas release flow diagram o f OGRES and TRAFFIC codes [from R . Cameron et. ai, Nucl. Energy, 29, 2, 1 47 (1990)].
bubble interlinkage. As can be seen from the overview of the mechanisms considered, shown in Fig. 2.12, the OGRES approach is somewhat simpler than that taken by GRASS. The FUTURE program1 07 is another well-known fuel performance code containing a fully mechanistic swelling and fission-gas release model. The fission gas release model may be said to follow the same general approach shown in the OGRES flow diagram (Fig. 2.12). However, the modeling of the grain boundary release process is somewhat more complex than that of OGRES. The complexity and long running time of fully mechanistic models, as well as the belief by some designers that simpler models are capable of adequate precision, has led to the use of simple fission gas release ap proaches in a number of fuel design programs. The simplest approaches are those based on empirical or semiempirical correlations. Some of the correlations proposed are as follows:
1 43
F U E L E L EM E NTS
given08 bysimple temperature zone model. The gas release fraction, isLewis1 (2.43) volumes of fuel in the temperature where VI'< 1000DC, V2 , V3 , V1 000 4 aretofractional regions 1 300DC, 1 300 to 1 600DC, and > 1 600DC, re spectively, and t is irradiation time in years. ' 2. Hoffman and Coplin s1 09 correlation between release fraction and av erage fuel temperature during maximum power. 11 0 3. Beyer and Hann temperature zone model. In this correlation, the gas release fraction, is given by (2.44) where Xl ' X2 , X3 are fractional amounts of gas produced between 1200 to 1400DC, 1 400 to 1 700DC, and > 1 700DC, respectively. Aschosen an alternative tosimple the simple correlation approach, some modelers have to develop models, which are deemed to cover the most significant of the gas release mechanisms. The probabilistic approach de I 1 assumes that (a) a proportion, K1 , of the fission veloped by Weisman et aI. I gas escapes directly from the atfuelgrainmatrixboundary withoutareas beingandtrapped and (b)is fraction of the gas trapped dislocations K2 released per unit time. The moles of gas released, r, at time t are given by r = p { t - 1K-I KK2 [ 1 exp( -K1 K2 t )] } (2.45) where p ' is the gas production rate and t is the time. assuming thatsteps, reactortotalpower operation can be describedTheby number a series ofof Bymoles constant power gas release can be determined. released, �ri' during the i 'th time interval, �ti' is then given by �r · = r·- 1 = �{ �t . - KI K2K1 [1 -exp( - KI K2�t. ) ]} +Ci_d1 -exp(- K1 K2�t)] (2.46) total gasis where is gastimeconcentration in thefuelfraction equal toofp'released t - Sincegas theproduced Cfrom ri' release zero is L� given by 1.
F,
F,
1
,
1
1
r 1· -
__
P1
1
-
1-
1
r
MasacDonald and Wang11 2 give constants Kl and KI K2 for 95% dense fuels
1 44
TH ERMAL A N AL Y S I S OF PWRS
K1 = exp( - 12450/ T + 1 .84)
t 2
(2.47)
K K = O.25 exp( - 21 41 0 / T )
1 where T i s i n o R and i s i n hours. The simple model of Verbeek and Hoppe47 approaches the problem by modifying the Booth diffusion equation 104 to consider both bubble resolu tion and gas trapped at defects or precipitated in bubbles. If capture and resolution rates are taken as constant, then the original Booth diffusion model can be used providing, the diffusion constant, which was orig inally is redefined as:
0' 0' 0"b/[a2(b
0/ a2,
=
+ g )]
(2.48)
where
0" o a b
= O + Kb Rf
original diffusion coefficient sphere radius = resolution rate = Ka Rf ' s Rf = fission rate density Ka , Kb = empirical constants g = capture rate, S - 1 =
=
To account for the enhanced fission gas release once bubble interlinkage begins, the sphere radius, a, is given one of two values. At low burnups, the sphere radius is considered to represent the actual size of the average particle making up the sintered compact. At high burnups, it is calculated as where is the measured fuel surface to volume ratio. Above the critical burnup, gas needs only to diffuse to the grain boundaries where interlinking bubbles are assumed to provide a path for its immediate re lease. The sphere radius is then taken as equal to the crystal grain radius. Since this is very much smaller than the particle size, release rates are con siderably enhanced . The critical burnup is related to the limiting porosity used in the swelling model of Eq. (2.37) . The sphere radius is decreased when
3/(s /v)
(s/v)
(2.49) where a is the existing fuel porosity and other symbols have the same meaning as in Eq. (2.37). The models of SPEAR-BETA 113 and ESCORE48 are a step closer to the fully mechanistic view. A small portion of the fission gas produced is re leased directly due to recoil. An additional direct release occurs in the grain growth region as small grains coalesce into the boundaries of larger grow-
F U E L E L EMENTS
1 45
ing grains. The bulk of the gas release is considered as indirect and deter mined by two steps. The first step is the release of the gas from within the grain to the grain boundary. The total fission gas concentration within a grain is determined by a balance between the production rate and losses due to recoil and transport to the grain boundary. Only gas that is mobile can be released. The ratio of mobile to trapped gas within the grain is determined by an Arrhenius-type reaction rate equation. The release rate can then be determined as the mobile gas concentration divided by a release rate time constant. The second step of the process is the release of gas from grain boundaries to the free volume. The rate depends on both the grain boundary concen tration and a second time constant. The differential equations48 are readily integrated and the integral equa tions are solved simultaneously for the changes in the concentration within the fuel grains and on the grain boundaries. For a given time step, the ESCORE model represents these by:
dCs = [[3Tg 0 - Pd ) - Cg,o ][ 1 -e - Mhg][1 -Bg ] - Cg,o Bg dCb = {[[30 - Pdhb -Cb,O ][ 1 - e - Mhb] + Tg� Tb [Cg,o - [3TgO - Pd )]
dt, (2.50) (2.5 1)
where
Cg,o = Cb,o = [3 = PdB == g Bb = Tg = Tb =
concentration in grain interiors at beginning of time step, moles per m 3 fuel volume concentration in grain boundaries at beginning of time step, moles per m3 fuel volume fission gas production rate, mol ! cm3 s direct release fraction (0.004) fraction of fission gas, from grain interiors immediately released by grain growth fraction of fission gas, from grain boundary immediately released by grain growth grain interior fractional release time constant, hr grain boundary fractional release time constant, hr.
The fraction of fission gas from within fuel grains that is transferred to the grain boundaries immediately upon grain growth, due to a fraction of the new grain volume being swept free of fission gas, is given by: (2.52)
T H E RMAL A NA L Y S I S OF PWR S
1 46
where
d
g,Q = average grain diameter at the beginning of the timestep, /-Lm = average grain diameter at the end of the timestep, /-Lm Bo = born-free exponent for the grain interior = 2 Bu = the burnup (MWd /MTU), with Bu set equal to 5000 for the range of a to 5000 MWd / MTU.
dg
Similarly the fraction of fission gas on grain boundaries that is released immediately upon grain growth is determined from (2.53) where Fo is the born-free exponent for the grain boundaries; it is equal to 3. In the equations for Bg and B/J above, the exponential involving burnup provides an incuba tion period that retards gas release at low burnups. The probable rate of fission gas release to the grain boundaries is as sumed to be directly proportional to Cg , mobile gas concentration within the grains, and inversely proportional to the time constant Tg . Since the release rate is highly temperature and grain size dependent, the time con stant governing this release is given by an Arrhenius type equation where
Tg
Tg
g R T Qg d R
= =
= = = =
= 0.895g [exp(Qg i / RT) - 45]
(2.54)
time constant for release from grain interiors, hr the mean grain size for each timestep, /-Lm gas constant = 1 .987 cal/mol K temperature, K the grain-interior activation energy (cal /mol) / R max{100 o r 7950[ 1 + 9 exp( - Bu /5000)]}
and Bu is defined and limited as for Eq. (2.52). The release from grain boundaries is considered to be proportional to the grain boundary concentration, C/J ' and inversely proportional to the grain boundary time constant, An Arrhenius rate constant equation is also used to compute
T/J. T/J
T/J.
= 5.09 X 1 0
- 17
[exp(Qgb / RT)]
where Tb
= time constant for release from grain boundaries, hr
R = gas constant = 1 .987 cal/ mol K - 1 T = temperature, K Qgb = grain-boundary activation energy (cal / mol) /R = max{100 or 6700[ 1 + 0.9 exp( - Bu /5000)]).
(2.55)
F U E L E L E M EN T S
1 47
The burnup, Bu, is again defined as for Eq. (2.52). The burnup term, ap pearing in the equations for the activation energies, models the decreased fission gas release seen at low burnups. The moles of gas released to the free volume, dCrel (mol/ m3 fuel), is determined by the balance between the production of fission gas and the changes in Cg and Cb during the timestep; that is, dCrel = dCp - dCs - dCb
(2.56)
where dCp is the fission gas produced in timestep (mol/ m3 fuel). The ESCORE fission gas release model has been shown to provide good agreement with a wide range of in-pile test data. The moderately sophisticated fission-gas release model by Rausch and Panisko 11 4 has been used as a conservative standard against which conser vative fuel rod design codes could be benchmarked. The modeling is di vided into high- and low-temperature submodels for both stable and radioactive isotopes. In the low-temperature model, the release of stable fission gas depends only on fuel burnup. In the high-temperature model, power and temperature are assumed to have remained constant for several half lives. The fractional release is taken as a function of time, fuel temper ature, and burnup. The fractional fission gas release for a particular ring of a given axial segment is computed as the larger of the values computed from the high- and low-temperature models. Since Pu is a major fraction of the fissile material in any fuel rod at the end of life, one would expect that models which are satisfactory at high burnup should apply to MOX fuel. However, release fractions from MOX fuel can be higher than those predicted by the usual models if there is substantial inhomogeneity. In an inhomogeneous fuel, the small granules of Pu02 experience a high local burnup. Since the fraction of fission gas release increases with burnup, fission gas releases will be higher than those expected on the basis of the average fuel burnup. 2 . 2 .4.2
Pressure L imits
Once the fission gas release has been obtained, the internal gas pressure,
Pi' may be determined from
where
M
pI. =
RM
----
L (V) T)
(2.57)
is the total moles of released fission gas and initial fill gas,
Li ( Vj ITj) is the sum of ratio of each free volume in rod to average absolute
R
temperature of the volume, and has its previous meaning of the ideal gas constant. The volumes available for gas containment include the open fuel poros ity, pellet-to-cladding gap, pellet cracks, dishes, and gas plenum. The com-
1 48
T H E RMAL A NA L Y S I S OF PWR S
putation of these volumes should be based on the hot fuel and cladding dimensions with appropriate allowances for creep and elastic deformation. In early rod-type fuel element designs, internal fission gas pressure at the end of life (EOL) was generally required to be no greater than the normal external system pressure. More recently, this criterion has been re laxed and EOL gas pressures are set so that there is no cladding "lift-off." Tha t is, once the cladding is finally in contact with the pellet, pellet-to-cladding gap is not allowed to develop. This assures that excessive fuel temperatures due to a large gas-filled gap cannot occur. The no lift-off requirement can be easily implemented in fuel rod design computations by requiring that the contact pressure be above a specified minimum value at full power at EOL. In fuel plates, unacceptable deformation of the plates limits fission gas release. Lustman,1 1 5 who assumed that plate compartments deform into trapezoidal prisms, obtained the following expressions for maximum de formation and internal pressure
y
Pi ( V -2ya2 N*RT P Y= a /3 o) P;=(N*RT)/(Vo+2ya2/3) Po Vo 0.01 38a4
Ec3
N*
(2.58) (2.59)
where is the moles of gas released, is the external pressure, c is the cladding thickness, is the initial free volume in plate element (cold con dition) (in. 3 ) and a is the edge length of plate compartment square (in.). Other symbols have their previous meanings. Creep has again been omitted. 2 .2 .4.3
Status of Fission Gas Release Models for Normal Opera tion
To obtain an appreciation of the magnitude of fission gas release from U02 pellets, we may make use of the approximation that the fractional fission gas release may be related to the volumetric average temperature of the fuel pellet at full power. l OY Experimental measurementsl 09 indicate that moderately burned U02 with average temperatures at or below 1 000°C re leases 5% or less of the fission gas produced . l o9 Release rates increase rap idly as the average temperature rises above 1 000°C. At a volumetric average temperature of 1400°C, moderately burned fuel will release about 30% of the gas produced. At a volumetric average temperature of 1 800°C, release rates of about 40% are seen at moderate burnups.1 09 During the late 1970s and beginning of the 1 980s, there was concern that available fission gas release models severely underpredicted gas releases at high burnup.l l 6 Some modelers moved to incorporate this. For example, earlier versions of the ESCORE model included a burnup enhancement fac tor. However, considerably more high burnup data has become available
F U E L E L EM E NTS
1 49
since these earlier observations. These data indicate that the rate of fission gas release does not accelerate at burnups up to 54,000 MWd /tonne U. 1 l 7 The current modeling approaches, without additional burnup enhancement terms, are now considered as being applicable to high-burnup fuel (e.g., the ESCORE model has been shown to apply up to 60 GWd/tonne). The specific details of the fission gas release process are likely to remain a matter of debate for a considerable period. Models that differ significantly in their details can often provide equally adequate data correlations. It is, however, generally recognized that (1) a small portion of the fission gas is released directly and the bulk of the fission gas is released indirectly through a diffusion and trapping process. Most mechanistic models also assume saturation of the traps at moderate burnup. Some early researchers concluded that fission gas release rates were en hanced during normal shutdown and startup periods. This effect does not seem to be specifically included in current models. The ability of various computer programs to predict fission gas release has been examined on several occasions. In a 1 983 study 107 of 16 European programs, most performed adequately for steady-state conditions but only the modified diffusion models of COMETHE III and FEMAX 1-111 1 18 gave satisfactory results for transients. Subsequently, the URANUS model was improved1 07 and all three of the models gave good results (e.g., prediction of �17% fractional release versus a 1 5% experimental value). Less satisfac tory models have sometimes predicted release rates that deviate from ex periment by a factor of However, the performance of the models cannot really be assessed independently since the other facets of the program strongly affect calculated releases. Totev and Dickson l 1 9 avoided this prob lem by examining the ability of the Rausch and Panisko, 114 Beyer and Hann, l 1 G and Weisman models using the FRAPCON program1 20 with the deformable pellet option for all the fission gas models. They found the Weisman model 1 1 1 to provide the best results with the discrepancy between observed and measured internal pressures being on the order of However, only six rods were considered and somewhat greater dis crepancies might have been seen if a larger sample had been used.
(2)
2.
3%.
2 . 2 .4.4
et al.
et al. 1 11
Fission Gas Release During Transients
In accident situations of very short duration (e.g., some design basis LOCAs where core uncovering may last only a few seconds), gas release during the transient is often ignored. Fission gas pressures are often computed using the gas inventory at the beginning of the accident and the fuel tem peratures of the accident period. However, for transients of longer duration the additional fission gas release during the accident could be significant. In the early 1980s, most of the computational models available at that time underpredicted transient fission gas release. The reasons for this un d erp rediction are not entirely clear. In some cases, the grain boundary
T H E RMAL A N A L Y S I S OF PWR S
1 50
sweeping of fission products, caused by rapid growth of equiaxed U02 grains at high temperature, may have been ignored by the computations. Macinnes and Gittus 1 2 1 suggest that the tiny gas bubbles within the fuel grains are not infinite sinks for gas atoms colliding with them during tran sien ts. By allowing only some of the colliding gas atoms to be trapped by the bubbles, Macinnes and Gittus were able to develop a revised gas release model that agreed with transient data. The fully mechanistic models of GRASS-ST, FASTGRASS, and FUTURE have been revised to take into account the available transient data. FUTURE has been shown to agree with the transient data that had previously given most programs difficulties. I ( )7 The ESCORE fission gas release model lEqs. to can also be applied to transient release. The model provides for direct release of fission gas via boundary sweeping [Eqs. and These release equations, coupled to an equiaxed grain growth model, would provide for rapid gas release at high temperatures. Leech et al. 1 22 feel that the most accurate results for transient release are obtained with an empirical approach. They suggest that the additional frac tional release obtained during a transient can be estimated from
(2.50) (2.56)] (2.53)].
(2.52)
F=
1
-
exp(
-
(2.60)
Kt )
where
T
K =
2 . 2 .5
temperature, K exp( time at elevated temperature (s).
48.3
-
23,450/T)
Thermal Resistance of Fuel and Fuel-Cladding Gap
(1994)
Since pressed and sintered oxide pellets are now used exclusively for cylindrical fuel rods, this section considers only this fuel fabrication approach. 2.2.5.1
Behavior with Open Gap at BOL
With pressed and sintered pellets, there is usually an appreciable resistance to heat transfer between the pellet surface and cladding. Since bonding agents are not used in PWR design, interfacial resistance can be that of a gas-filled gap or U02 in actual contact with the cladding surface. It has often been assumed that at the beginning of life (BOL) the pellet remains nearly centered within the rod and that gap conductance can be obtained from the conductance of an annular gas gap. If this assumption is correct, then gap conductance hf can be estimated from h = t l + lo
kg
(2.61 )
F U E L E L EM E NTS
where kg is the thermal conductivity of gas in gap (Btu / hOF ft), I is the gap width at operating conditions, and 1 0 is the constant specified to provide a cond uctance consistent with that computed at zero contact pressure. Cohen et a1.1 23 investigated gap conductance for U02 stainless-steel-clad fuel elements under operating fluxes. The operating gap was obtained by correcting for the thermal expansion of cladding and fuel and by assuming the fuel remains centered within the cladding. Figure shows that data can be fitted by an equation like Eq. Although radiation from the pellet to the cladding would very slightly increase the gap conductance, this effect is not normally considered. If a 3-mil radial gap filled with helium is assumed for a standard fuel rod operating kW ft, radiation would increase the gap conductance by only about at With smaller size BOL gaps (which are usual) or pellet-cladding con tact, the effect of radiation heat transfer would be even smaller. Proper evaluation of the gap conductance, prior to the establishment of contact between the cladding and fuel pellet, depends on determining ap propriate values for the gas thermal conductivity, kg , and the parameter 10 . Gas thermal conductivity can be determined from the data of Von Ubisch et ai.,1 24 who examined the thermal conductivity of binary and ternary mixtures of xenon, krypton, and helium. They concluded that a mixture of xenon and krypton, which approximates fission gas Kr, Xe), had a thermal conductivity, kg , represented by
(2.61).
2. 1 3
1302)5
(15.3% 84.7% (2.62)
2.5%.16 1
kg = ko (T
where T = gas temperature, K ko = s
=
147 0.86
kg = gas thermal conductivity,
10-7
cal/cm2 K s.
Since helium is used to pressurize the fuel rods initially, its conductivity is required. The thermal conductivity of helium can be represented by Eq. with ko = and s = To estimate the thermal conductivity of mixtures of fission gas and helium, Von Ubish et af.1 24 suggest
(2.62)
3670
0.72.
(2.63)
wh ere
km = thermal conductivity of mixture kHe = thermal conductivity of helium kFG = thermal conductivity of fission gas x = mole fraction helium.
T H E RMAL A N A L Y S I S OF PWR S
�oo 300
Symbol
1 00
X
70
50
I
N E u
22-2
0
22-9
� Q)"
u c 0
8
7 .5
22-1
V
22- 1 4
®
22-5
•
5
8
6
•
7
13
22-3
0
1 0.0
Initial Gap, m i l
22-7
0
•
30
�
Exp. WAPD No .
3 .5
22- 1 5
3 .5
22-4
1 .5
2.5
1 .5
22- 1 1
0
3
U
::J " C 0
U
a. 0
1 .0
()
0.7 0.5 0.3
X )0( XX
O cP' cf
0. 1
X X
X )O(
0 .07 0 . 05 0 . 03
. 01
o
2
3
4
5
6
7
Opera t i ng Gap, i n .
Fig.
2.1 3
8 x
9
1 0-3
10
11
12
13
Gap thermal conductance versus operating gap; fill gas-air at atmospheric pressure [from Ref. 1 23].
F U E L E L EM E NTS
Although a more complex approach, based on the kinetic theory of gases, is more commonly used for evaluation of mixture conductivities, it is not clear that this is warranted in view of the other uncertainties in determining the gap conductance. As long as the effective gap thickness is relatively large with respect to the mean free path of the gas molecules (from about to em), the gas thermal conductively may be taken a s independent of pressure. Standard PWRs are designed so that the cladding is "free standing" and, hence, the cladding will not collapse on the fuel pellet. The tolerance re quired to add pellets to a metal tube without problems are such that there is a pellet to cladding gap at initial operating conditions. However, pres surized heavy water reactors, such as the CANDU design, using natural uranium have little excess reactivity. It is therefore desirable to minimize parasitic absorption by using thin cladding, which collapses onto the sur face of the fuel. The pressure difference �P, which produces such a collapse, may be conservatively estimated from1 25
2x l O- 4 5X10 -5
�p
E l ( Lc) 4(1-' ) R
3
(2.64)
where
E
Young's modulus (evaluated at collapse stress) Lc = cladding thickness R = mean radius of cladding tube 'Y = Poisson's ratio (evaluated at collapse stress). =
(2.64),
When operating below the differential pressure indicated by Eq. a gap may be expected between pellet and cladding at BOL operating conditions. The determination of the nominal gap thickness at operating conditions requires the proper determination of the pellet and fuel diameters. Pellet diameters must be corrected for thermal expansion, net irradiation swelling, and if fuel temperatures are high, for creep and hot pressing. The cladding dia meter must be determined by allowing for thermal expansion, creep, and for the elastic deformation of the cladding due to the differential be tween external and internal pressure. For an infinitely long cylinder, subject to internal and external pressure, elastic deformation, � D i' is given by
(2.65) where
Di = internal diameter, in. Do = external diameter, in.
T H E RMAL A NA L Y S I S OF PWRS
1 54
Pi = Po =
E
K 'Y =
internal pressure, Ib/in. 2 external pressure, Ib /in. 2 Young' s modulus for the cladding Do /Dj Poisson' s ratio for the cladding.
Once the gas thermal conductivity and nominal gas thickness are known, the gap conductance may be calculated if an estimate of the parameter 10 in Eq. is available. This parameter allows for the effects of surface roughness and the lack of thermal equilibrium between the gas molecules and solid surface in a confined space. The parameter may be obtained from the sum 126
(2.61)
(2.66)
where
!3' , g l g2 g3 1 1 ,1 2
=
=
(2.0
120)
empirical constant according to Ref. arthmetic mean roughness of cladding and fuel, respectively, m fragmented fuel roughness, m temperature jump distances at fuel and cladding, respectively.
The nominal gap may be thought of as the distance between the high points on the rough fuel and pellet surfaces. Since the surfaces are not flat, a portion of these surfaces is separated by distance greater than the nominal gap. The product of f3' and the roughness factors, g l and g2 ' thus increases the effective distance between the two surfaces to account for this fact. Some investigators add the additional term g3 to account for the additional fuel roughness introduced by in-pile operation. Values between X and m have been suggested for g3 (Ref. It is found 1 27 that at gap distances having an order of magnitude not far from that of the mean free path of the gas molecules, the energy interchange between gas molecules and a solid surface is incomplete. The accommo dation coefficient, (x, is defined as the ratio of the temperature difference between the incident and reflected gas molecules to the temperature dif ference between the solid and incident molecules. The temperature jump distance required by Eq. may be computed from the accommodation coefficient by 1 27
8x10-6
48).
5 10- 6
(2.66)
(2.67)
where k
=
P
=
thermal conductivity of gas molecule, W /m K gas pressure, Pa
F U E L E L EM E NTS
T = average gas temperature, K a j = accommodation coefficient of gas component i Mj = molecular weight of i'th component, kg/ mol Xj = mole fraction of the i'th component. Lanning and Hann 128 suggest the accommodation coefficient can be taken as a property of fill gas only. Based on the data of Ullman et al.} 29 they suggest a i-Ie =
�
a e=
0.425 - 2.3
(2.68) (2.69)
X 10 - 4 Ts
0.749 - 2.5 X 1 0 - 4 Ts
where Ts is the surface temperature (K). Accommodation coefficients for other gases are obtained by linear interpolation between ai-Ie and a � e based on molecular weight. Temperature jump distances for a gas mixture may also be estimated directly from the temperature jump distances of pure gases. Ross and Stoute 130 estimated these quantities for helium, argon, and fission gases from their data. For temperatures between 1 50 and 00°C, (1 1 + l2 ) is < 1 0 X 1 0 - 4 cm for helium, 5 X 1 0 - 4 cm for argon, and < 1 X 1 0 - 4 cm for xe non. The distances are assumed proportional to the mfp of the gas mole cules and, hence, inversely proportional to the temperature. Campbell and DesHaies13 1 verified Ross and Stoute's temperature jump distance esti mates. To use these data for gas mixtures, Veckerman and Harris132 show that
3
1m =
L Xj/j / (M)l12 ----
L xj /(Mj)V2
(2.70)
where
1m = I; = Xj = Mj =
temperature jump distance of the gas mixture temperature jump distance of constituent gases mole fraction molecular weight.
Wesley and Yovanovich133 propose a substantially different approach for the determination of the gap conductance, hf . They consider the stochastic variation in the gap size due to the variation of the heights of the projections on the two gap surfaces. The gap is described in terms of Y, the distance between the surface mean planes, and an effective surface roughness, u . If it is assumed that the gas mixture in the gap can be treated as a single gas with appropriately averaged properties, then Wesley and Yovanovich's re sult may be approximated as
T H E RMAL A N A LY S I S OF PWRS
1 56
3.0
n
= 10
2.5
n
=10
-3
2.0 =10
RI
Q
1 .5
1 .0
O.
Fig. 2 . 1 4
-2
'" 2
3
= 1 0
Q
n 4
= 1
=10
6
5
7
Y/ a
8
9
��
10
Roughness factor versus dimensionless temperature jump lengths [from Ref. 1 33] .
(2.71) where
a = (af + ai)ll2 a1 ,(J'2 = = = (1 +i2 )/ a . 2. 1 4 R1
n
standard deviation of surface projection heights of cladding and fuel, respectively. roughness factor determined from Fig.
(2.14)
Y / a >5. (2.6 1) (2.66)].
(2.71)
Figure shows that roughness effects die out rapidly with increasing gap spacing and that they can be ignored for Equation would seem to provide a more exact calculation of hf when gap sizes are small. Here conductances are considerably above those calculated by the more conventional approach [Eqs. and This is due to contact be tween the larger projections even though there is still a nominal gap. Postirradiation examination of fuel rods with an open gap indicates that the fuel temperature predictions are somewha t too high when gap conduc-
F U E L E L EM E NTS
(2.61)
and the assumption of a conventional tance models are based on Eq. 1 annular gap. 34 When a simple annular gap model is used, the centric con for helium-filled rods operating gap thickness needs to be reduced by about for pure xenon to produce best estimate temperature pre nd a dictions.48 It has been postulated that the reduced temperatures were due to the fuel pellet columns being nonconcentric and in partial contact with the cladding. An alternative explanation is that pellet cracking produced partial pellet-cladding contact (see Section
75%
�25%
2.2.5.2).
2.2.5.2
Performance with Closed Gap
When the pellet expands sufficiently to be in direct contact with the clad ding, an improved gap conductance results. Several theoretical explanations of the phenomenon of contact conductance have been proposed and, in general, they consider a model in which the solid surfaces are in actual contact at only a few discrete points (Fig. Since the thermal conduc tivity of the solids in contact is generally much greater than the thermal conductivity of gas filling with interstices, heat flow tends to channel through the points of contact. When contact pressure is increased, peaks in contact are deformed and the contact points increase in size and number. The number and size of the contact points also depend on the surface fin ishes. At low contact pressures, the primary mode of heat transfer can be by conduction across the gas gaps. The effective distance between the sur faces depends on the size and shape of surface irregularities. Various analytical approaches have been proposed for computing con tact conductance. Cetinkale and Fishenden1 35 presented an equation for the contact conductance of two metal surfaces based on a calculation of the isotherms at the contact points of the two solids. A somewhat different approach was taken by Fenech and Rohsenow,1 36 who found an approxi mate analytical solution for heat conduction at a contact point. The results obtained from these approaches have not been widely used by designers. A revised form of the Mikic and Todreas1 37 model was used in fuel rod design but this is now superseded.
2. 1 5).
Fig. 2 . 1 5
Contact conductance model .
T H E RMAL A N A L Y S I S OF PWRS
1 58
The total gap conductance, hg , is often computed following the method suggested by Ross and Stoute.1 30 The conductance is taken as the sum of hs, the conductance due to solid contact regions, and h , the gas conductance f h = h s + hf g
(2.61)
(2.72)
One approach to the evaluation of hf would be the use of Eq. with the assumption that the nominal gap, I, is zero and that 10 can be obtained by the procedures of the previous section. This, however, ignores the de crease in surface projection heights due to externally applied pressure. Lan ning and Hann 1 28 conclude that the gas conductance can be obtained from
(2.73)
where � L eff =
K1 K2 = C =
K 1 [ C (g l + g2 ) + (l 1 + I2)] + K2
1.8 -0.00012 2 0.000125 P)
exp( contact pressure, kg/ cm2 g l ,g2 = mean surface roughness (projection height) of fuel and cladding, respectively, cm 1 ,12 = temperature jump distance, cm. 1 The contact conductance, h Sf in Eq. corresponds to the solid conduc tance that would be observed in a vacuum. It is usually evaluated by means of empirical correlations of data that have been corrected for the effect of gas conductance. Although some data on irradiated specimens is availa ble,1 23 these data tend to be clouded by the effects of irradiation on U02 properties. The correlations therefore are based on out-of-pile data. These data include the very early data of Wheeler 1 38(see Fig. which showed the importance of the yield strength in determining the conductance. More recent data include those of Dean, 139 Ross and Stoute, 130 Campbell, 140 Gar nier and Begej, 14 1 and Madhusudana. 142 Although general approaches to contact conductance are available, most modelers have simply correlated the Zircaloy-U02 contact conductance data. The simplest approach that has been used is to consider the contact conductance as directly proportional to the contact pressure
P
(2.72)
2.1 6),
P
(in vacuo 1.36 10- 3
(2.74)
where is the contact pressure (Pa) and a is the empirical constant. The ESCORE program48 states a as X W / (m2 °C Pa). The most reliable data) of the available experimental data U02Zircaloy contact conductance indicate that a less than linear dependence is appropriate. Madhusudana and Fletcher143 correlated these data by
F U E L E L EM E N TS
1 59
1 00 000
3 1 .6
1 0 000
3.16
E
::J ::J U 0
> c
Q) u C 0
t
::J -u c 0
U
.E
-
� 0
N
E
�
co
u
1 000
+-
u c
8
� 0.3 1 6 �
N
.t ......
G 0
...£:
S.S.-U02 1 00 0.000 1
-
+
-+
Zr-2U02
0.00 1
0. 1
0.0 1
0.03 1 6 1 .0
Contoct Pressure Y i e l d Strength Fig. 2.1 6
Contact conductance measured by Wheeler [from Reactor Handbook, Vol. IV, Interscience Publishers, Inc., New York (1 964) J .
h
L
� = 12.3 X 1 0 - 3 km
(�) 0.66 H
(2.75)
where ( g1 �) 1/2 = root mean surface roughness rms roughness of fuel and cladding, respectively gl,g2 == contact pressure P H' = flow pressure (Meyer hardness) of softer material km = harmonic mean of thermal conductivities k l and k2 of contacting surfaces [(2k1 k2) / (k1 k2)]. Typical values of H' are 13 X 1 04 psi for stainless steel and 14.2 X 1 04 psi for Zircaloy. 26 correlate essentially the same set of data with Lassman and Hohlefeld1 an equation of the form +
Lo
+
(2.76)
where a' = empirical constant g = arithmetic mean roughness
1 60
T H E RMAL A N A L Y S I S OF PWRS
k ITy
/2 (2.74) (2.75) (2. 74).(2.74)
(k 1 + k2 ) = yield stress of softer material. =
Both Eqs. and appear to model the reliable U02-Zircaloy data adequately. They both predict similar effects for contact pressure, hardness, or yield and thermal conductivity. However, the effect of surface roughness is more pronounced in Eq. Equation may be used for both stain less steel and Zircaloy but Eq. was derived for Zircaloy only. General correlations of contact conductance that can be used with any combination of surfaces are available. The best of these is probably that of Shylkov.144 This correlation appears to give results that are somewhat above most of the U02-Zircaloy data. 1 28 There now appears to be no advantage in the use of generalized correlations for fuel rod calculations. Contact pressure between the pellet and cladding can be estimated by computing their dimensions while ignoring any elastic deformation of the cladding. Once contact is established, the unrestrained pellet diameter will exceed the unrestrained internal cladding diameter. When the fuel operates at the relatively low linear power of current (1995) PWR designs, the pellet is essentially rigid. Pellet deformation occurs due to thermal expansion, swelling, and densification. The displacement of the fuel is then indepen dent of the cladding deformation. With the rigid pellet assumption, the internal pressure required to produce an elastic cladding expansion that would provide a zero gap is given by Eq. Since the cladding and pellet are in true contact over only a small fraction of the total area, the internal pressure, Pi ' is given by
(2.75)
(2.65).
(2.77)
At high fuel temperatures (e.g., accident situations) the rigid pellet as sumption is not entirely appropriate. Fuel deformation under stress (hot pressing) should be considered and deformation of fuel and cladding must be computed simultaneously. If the fuel is operated with pellet contact and nearly constant power for an appreciable period, the cladding creep rate will be in equilibrium with the net swelling rate of the fuel. The cladding circumferential stress re quired to produce this creep rate can be determined from an appropriate creep rate equation. This stress can, in turn, be used to estimate the internal pressure and from that the contact pressure. By assuming that the thin wall approximation holds,
(2.78)
where cladding thickness o = average diameter of cladding
Lc
=
1 61
F U EL E L EM E NTS
Pi,Po = internal and external pressure, respectively all
=
2 .2.5.3
circumferential cladding stress.
Performance With Open Cap and Cracked Pellet
Since U02 cracks so extensively at the heat generation levels at which a PWR operates, it has been suggested that it is unrealistic to assume that an annular gas gap can be retained after a few hundred hours of operation, even if calculations indicate there is a significant space between pellet and cladding. It is likely that cracked pellet pieces are lying against the cladding so gap conductance would approach the value computed from one of the contact conductance equations while assuming a zero contact pressure. The data of Calza-Bini et support this point of view. They observed that on the first rise to power, Eq. provided reasonable estimates of the measured gap conductance. However, on subsequent power cycles, the measured gap conductance increased significantly. Stehle et report that Brzoka and Wunderlich collected results showing somewhat similar behav ior. They concluded that of the initial gap was eliminated during the first rise to power. After hr of irradiation, they concluded that another of the gap was closed. These estimates apparently made no allow ance for a decrease in U02 thermal conductivity due to cracking. It is likely that such an allowance would lead to estimates of nearly complete gap closure after hr of operation. Lanning et avoided the assumption of an annular gap and simply correlated in-reactor data from sources. For helium-filled rods, they suggest
al. 1 45 (2.61)
al?6
�40% 500
�20%
500al. 1 46
16 VTgap - 2.36 hg =0.0321 + 0. 1 167 2L' /D 2L'D hg - 0. 1 129 + 2L'0. 1 401/D + 0.0069q' D
--
(2.79)
For krypton- and xenon-filled rods, _
(2.80)
where
Tgap q' L' D hg
gap temperature, K linear heat rating, W / em = gap thickness = tube diameter and is given in W cm2 DC. MacDonald and Weisman46 proposed a fairly complete model for cra cked pellet behavior. They conclude that after a short period of initial operation at power, the pellet cracks and segments move outward, lying =
=
/
T H E RMAL A N A L Y S I S OF PWRS
1 62
against the cladding. This causes an increase in gap conductance, but a decrease in fuel conductivity. They indicate that when the uncracked pellet diameter is less than the cladding diameter, gap conductance can be rep resented by
fk -f)-k-ghg =�+(1 L' L' +L'
(2.8 1)
0
o
where
f Lb = L' = =
fraction o f the pellet surface i n contact with the cladding a t zero contact pressure root mean square of fuel and cladding surface projections gap thickness neglecting pellet cracking or eccentricity.
Based on Kjaerheim and Rolstaad ' s suggestion, 147 the portion of fuel in contact with the cladding is obtained from C 1 ) CiL'C3 / D) C1 +
(2.82)
f = (1 -
where
C1 = = D= 2 = 0.2, C2,C
3
fraction of cracked fuel pellet assumed to be in contact with the cladding, which is independent of gap size constants representing the rate at which contact area fraction decreases with increasing gap hot pellet diameter.
(L' <0),
= 0.3,
1 ( l
(2.83)
For the data examined, MacDonald and Weisman concluded C1 C and C3 Once the gap closed gap conductance was represented by a simplified Ross and Stoute 130 model. By assuming that most of the annular gap is replaced by crack area in the pellet, MacDonald and Weisman46 derived an effective thermal con ductivity, for the cracked pellet. They obtained
= 50.
ke'
ke =kUO
2
where
kU0 2 = kg D L' =
1 2L' D[B(kg / kuo) A
+C]
thermal conductivity o f uncracked thermal conductivity of gas diameter of fuel pellet.
+
U02
2.5, B 0.077, D C = 0. 0 15. 2L' �2.5, ke=ku0 2.
With as gap thickness in mils and expressed in mils, they found that the constants were A = = and For
1 63
F U E L E L EM E N TS
MacDonald and Weisman46 found that the centerline fuel temperatures, calculated using their cracked pellet model, compared very well with cen terline temperature measurements in several instrumented fuel rod irradiations. More recently MacDonald and Smith148 analyzed a series of in-pile ex periments in which they examined the effects of fuel cracking on fuel con ductivity. They concluded that the cracking leads to an outward fuel relocation such that firm pellet-cladding contact occurs sooner than ex pected. They also concluded that the relocation for l -cm-diameter fuel pel lets operating at LWR power levels could be expressed as
Rp = 0.014 0.1 4L; +
Rp
L;
(2.84)
ke'
where is the outward pellet relocation (mm) and is the initial cold gap (mm). They then expressed the effective thermal conductivity, of the fuel in terms of a hot gap corrected for the relocation. For low-burnup rods containing helium as the fuel gas,
ke = kU02 -(0.0002189 -0.050867\fJ 5.6578\fJ2) \fJ = (L' -Rp)(0.0545 L�) L' = U02, kU0 2 = (2.85) (2.85), L' = Rp +
(2.85)
where
I hot gap without considering relocation, mm thermal conductivity of kW I m K.
Although Eq. is based on helium fill gas, it should be applicable in the early portion of fuel life where helium is the main fuel gas constituent and cracking most significantly affects fuel properties. Note that according to Eq. the fuel conductivity is essentially that of un when cracked fuel. Later studies48 of irradiated rods with significant burnups indicated that the average outward relocation of the cracked fuel is a function of burnup, linear power rating, rod diameter, and gap size. Krammen and Freeburn48 suggest that the average relocation, can be obtained from
Rp' o Rp Do = 0.80[Q(Lt Do)(0.005Bu .3 -0.20 Do 0.3)] 1i3 q6' !S6 Q= {o -_6)10)/2 q ' >14 Do I
w here
I
(q' (q '
and where
q'
Bu
=
=
+
for for for
kW I ft kW Ift
the as-fabricated cold diameter of the pellet, in.
the local linear heat rating, kW I ft the local fuel burnup, MWd /MTU
(2.86)
T H E RMAL A N A L Y S I S OF PWR S
1 64
Lt
L
O Lc Iosllre
= LO - L closu re = the as-fabricated cold diametral gap, in. = the permanent diametral gap closure that has already occurred, in.
Krammen and Freeburn indicate that after pellet contact is established, most of the pellet relocation is recovered over a period of time. They con clude that the recoverable relocation is limited to a 0.1 decrease in the pellet diameter. Although no relationship for relocation recovery is avail able in the open literature it is clear that the contact pressure must remain low until all, or the maximum allowable, relocation is recovered. The assumption of nominal low-contact pressure, which allows continued clad creep-down until closure, would be expected to produce satisfactory fuel temperatures. When Krammen and Freeburn estimate gap conductance, they add the value of Rp to the hot pellet diameter in order to calculate the gap thickness. Although they make no direct allowance for any pellet to cladding contact until the corrected pellet diameter equals the internal cladding diameter, they employed a further gap reduction which they state accounts for pellet eccentricity. It would appear that both the pellet relocation concept and the cracked pellet model are two somewhat different approaches to modeling of the same phenomenon. Acceptable fuel temperature estimates appear to be attainable by either approach.
(992)
2 . 2 .6
%
Mechanical Behavior of Cladd ing
Calculation of the fuel-cladding interactions depends on an accurate de scription of both cladding and fuel behavior. Fuel pellets expand thermally and swell (possibly densify) and when subject to pressure by the cladding, creep. Cladding expands thermally, deforms elastically, and creeps inward or outward depending on the direction of pressure forces. The hexagonal close-packed lattice of a-Zircaloy exhibits a preferred crystallographic orientation that confers a pronounced anistropy in some mechanical properties. The thermal expansion of a-Zircaloy is one of the properties that differs along the various crystal planes. Further, the drawing operation to which Zircaloy tubing is subjected results in a preferred crystal orientation. A fully accurate determination of the thermal expansion re quires 0 ) a determination of the thermal expansion of single crystals in the direction parallel and perpendicular to the basal crystal plane and a knowledge of the crystal or grain orientations in the sample of interest. The crystal orientation in a sample is referred to as the sample It is obtained by means of a To construct a pole figure, a sample of the material of interest is consid ered as the center of a sphere. Each of the crystal planes of the single crystals
pole figure.
texture.
(2)
1 65
F U E L E L EM E NTS
making up the sample will have a projection on the surface of the sphere. The further projection of the sphere surface locations onto a plane tangent to the sphere is the pole figure. If the crystals are all aligned in one direction, a set of single dots will appear on the figure. In the real case, clusters of dots will appear. These clusters may be used to obtain the average orientations of the crystals. By an appropriate volume weighting over the sample, the average sample properties may be obtained from the single crystal proper ties. When this detailed texture information is available, highly accurate clad ding thermal expansions can be computed (see Hagrman et al 34). When detailed texture information is lacking, the following correlations represent reasonable approximations of typical Zircaloy cladding behav ior. 34 For diametral change
.
-2.344 10 -4 + 10 - 6 T - 2.788xI0- 5 +4.44xl0-6 T T (300 < T < 1073 1090 1230 1230 10-6 u, {(1.9.20188 101011 1-4.1 -5.05475107T107T + K1 ) K2 88 + X
dD/ D =
6.72 X
(2.87a)
and for axial changes
(2.87b)
dL / L =
when (d D / D) and (d L / L) represent the total change in length per unit length and is in K. The foregoing holds for K). As previously noted, the Zircaloy alpha-beta phase transition starts at � K and ends at K for oxygen-free cladding. Above K both axial and radial expansion of beta-Zircaloy can be taken as �9.7 x in. l in. K. At low loads, Zircaloy strain, £, is proportional to the stress, divided by E, Young's modulus. Hagrman et aP4 give Young' s modulus as �
E=
X X X X linear interpolation
/
(3
ex
ex
phase phase (3 phase
(2 . )
where E =
T Kl K2 = C =
=
=
4> t =
Young's modulus for Zr - 2 and 4 with random texture, Pa cladding temperature, K cold work effect term = - 2.6 X C, Pa fast neutron fluence effect term = 0.12(4)t / cold work area ratio fast neutron fluence, n / m -
C= {o 0
1010 0.88 +
2
1025)
The cold work area ratio is estimated from the yield strength by: - . 269
1
+0 0019 3u , .
2
y
,
U140
(2.89)
T H E RMAL A N A L Y S I S OF PWRS
1 66
et al.
where ITy is the as-fabricated room-temperature axial yield strength (MPa). Hagrman also include a term for the effect of as-fabricated oxygen concentration. However, these concentrations are so low that this effect is negligible and therefore this term has been omitted here. Plastic deformation of Zircaloy can be estimated from ( £ = 0" / K ) l / n
(2.90)
where K is a strength coefficient and n is a dimensionless strain-hardening coefficient. Values for these quantities from Busby' s 149 data are n = - 1 .3498 X 1 0 - 2 + 5.152 X 1 0 - 4 T - 5.5947 X 1 0 - 7 T 2
(2.91 a)
K = 1 .398304 X 1 0 19 - 1 .219965 X 1 06T - 96.6206T 2 [N /m2 ]
(2.91b)
Temperature T is in K. More recent data show that the parameters K and n are functions of the cold work and irradiation the tube has received.34 Transition from the elastic to plastic region occurs when plastic strain is greater than elastic strain. Neutron irradiation has a substantial effect on the strength and ductility of Zircaloy. Irradiation leads to increases in the yield strength and decreases in ductility. Because of irradiation-induced ductility reduction and further possible reductions due to hydride formation, the maximum permissible strain must be set at a low level. Some irradiation experience150 indicates that the maximum strain should not exceed �1 since some test rods have failed at strains of this magnitude. Many investigators consider the strain range (maximum outward £- and minimum inward £), to be a better cri terion for brittle failure. It was previously observed that fast neutron irradiation also effects Zir caloy creep. These effects have been investigated by Ibrahim1 51 and others. Early correlations generally gave the creep rate as the product of neutron flux term and a stress term. For example,151
%,
£ = K [O" + B exp(CO")]exp( - 1 0000/
where
RDt-
(2.92)
£ = transverse creep rate, m/ m s K = 5.129 X 10 - 29 C = 4.697 X 1 0 - 8 T = temperature, K = fast neutron flux, n / m2 s (£ > 1 .0 MeV)
B = 7.52 X 1 02 = 1 .987 cal/mol = time, s .
R t
K
More recent representations of Zircaloy creep express the creep as a sum of thermal and irradiation creep terms. That is:
1 67
F U E L E L EM E NTS
(2.93) E total = E thermal + E irradiation Based on the thermal creep-model of Gorsuk and Pfennigworth1 52 and the irradiation-creep model of Franklin 1 53, Krammen and Freeburn48 reco mmend:
E thermal = A l [sinh(A2 a :3 )] A 4 tAS a:6 exp( - A
IT)
(2.94)
7 Eirradiation = B l t B2 B3a :4 [exp( - B5 T ) ]a�6 [cosema x ] B7
I
(2.95)
where
E = total diametral creep strain, m/m t = exposure time, hr = fast neutron flux for E >l MeV, n / cm2 · s all = midwall hoop tangential stress, MPa T = cladding average temperature, K ay = cladding yield strength at room temperature, MPa cos8 max = cosine of angle of maximum intensity of basal pole to radial direction, normal to axial direction (obtained from pole figure). The constants in the creep equations, which were derived from a regression fit of LWR creep rate data, have the following values: Constant Al A2 A3
A4 As A6 A7
Value 1 .388 X 1 08 3.29 X 10 - s 2.28 0.997 0.770 0.956 23 X 1 03
Constant Bl B2 B3
B4 Bs
B6 B7
Value 21
2.35 1 0 0.81 1 0.595 1 .352 22.91 1 .58 2.228
Note that the Eq. (2.95) requires Zircaloy texture information. When this is not available, Eq. (2.92) can be used to provide a reasonable estimate of average cladding behavior. The creep strain correlation of Eqs. (2.94) and (2.95) is based on data from both compressive stresses (seen in creep-down when external pressure exceeds internal) and tensile stress (seen in outward creep when internal pressure exceeds external). However, some investigators use separate cor relations for compressive and tensile creep. 34 The predictions of total creep of Eqs. (2.94) and (2.95) and the creep rate of Eq. (2.92) are based on the data taken with the Zircaloy subjected to constant stress from zero time until time t. In any reactor situation, the stress will vary with time and it is not obvious how the correlations should be applied . Two approaches have been proposed : (1 ) The equivalent creep rate is a function only of the time over which stress
Time hardening law:
T H E RMAL A N A L Y S I S OF PWRS
1 68
Strain hardening law:
The equivalent creep rate is only has been applied. (2) a function of the total strain. Current theory holds that the strain hardening law is the most appropriate. To determine the creep, over a given time interval with a specified stress and initial strain, with this approach, an equivalent time, is required. This time is that needed to produce the specified strain if the current stress had been a constant from time zero. This equivalent time may then be used directly in Eq. (2.92) to determine a strain rate. Alternatively, the strain rate may be determined by substitu tion of teg into the partial derivatives of Eqs. (2.94) and (2.95) taken with respect to time. Irradiation-included axial growth of Zircaloy can be of significant con cern for high-burnup fuels. A fairly complex model is provided in the later version of MA TPRO.34 A simple correlation is provided by Hagrman 1 54 as follows:
teg,
(2.96) where
�L / L A
To T
4> t Cw
13
fractional change in length (neutrons /m2 ) - 1 / 2 240.8 K = cladding temperature, K = fast neutron flux, n/m2 s (E > 1 .0 MeV) = time, s = degree of cold work (fraction of cross-sectional area reduction) texture factor (0.05 is a typical value). =
1 .407 x 1 0 - 6
It has been observed that axial growth depends on the tubing supplier. Some designers48 therefore prefer to correlate these data by
� L / L = A(4) 't)r7
(2.9 7)
where 4> ' is in n / cm2s and A and n are determined for the given supplier. Values of A vary from 1 .8 X l 0 - 15 to 7 X l 0 - 2o while varies from about 0.55 to 1 .0 .
n
2.2.7
Accuracy o f Fuel Element Property a n d Behavior Models
Those models or correlations describing simple fuel and cladding properties (e.g., coefficient of thermal expansion, specific heat, density) generally ex hibit good accuracy. Unless noted to the contrary, the properties are as sumed to be unaffected by fuel or cladding irradiation. When compared to experimental measurements at the high temperatures encountered at reac tor conditions, most properties of unirradiated materials show root mean
F U E L E L EM E NTS
1 69
square percentage errors (standard error) of about 10% (Ref. 34). The gen eral difficulty of high-temperature measurements accounts for most of this error. Variation in sample preparation is another contributor to the error. The effect of irradiation on fuel and cladding properties introduces ad ditional uncertainties. For most properties, irradiation effects are entirely negligible or very small as noted earlier. Irradia tion has a small effect on the melting point of oxide fuels.50 The effect of irradiation on oxide fuel thermal conductivity is of considerably more importance to the designer. Because of the structural changes that oxide fuel undergoes during irradi ation, it is difficult to disentangle the irradiation effect on the conductivity of the oxide itself from the effects of fuel cracking and changes in gap conductance. Estimates of the change in oxide conductivity due to irradi ation may therefore show errors of ± 50% . However, since the irradiation effect reduces thermal conductivity by no more than about 15% (Ref. 48), the effect of this error on temperature measurements is modest. Models and correlations for cladding and oxide fuel behavior (e.g., clad ding creep rate, fuel hot pressing, fuel cracking, gap conductance) generally have a lower accuracy than those for properties. Cladding behavior can be significantly influenced by the exact fabrication method employed since this will vary the cladding texture. Oxide fuel behavior can be strongly influ enced by the properties of the oxide powders used to produce the pellet. Information on cladding texture or oxide powder properties is often lacking and most general models therefore ignore these factors. In addition, irra diation effects are usually significant. Since determining the irradiation the sample has received is difficult, a further uncertainty is introduced. The problem is exacerbated by the possibility that the form of the model or correlation may not be entirely appropriate for the phenomena involved. Fuel and cladding behavior models will generally compare reasonably well with the data set used to generate the models. However, when com pared to other data sets, errors of up to 50% may sometimes be encoun tered.53 Despite the uncertainties in property and behavior correlations, predictions of key thermal parameters of operating fuel rods generally show reasonable agreement with experimental data. For example, predic tionsS3 of centerline fuel temperatures as a function of power in a typical PWR fuel rod with low burnup tend to agree with actual measurements to within about 200°F. 2.3
STEA DY-STATE CO N D UCTION U N D E R I D EAL IZED CON D I T I O N S
2.3.1 2 . 3 . 1 .1
Heat Generation With in a Fuel Plate Uniform Heat Generation
Let us consider the temperature distribution within the fuel of a thin fuel plate. If we ignore axial conduction, a one-dimensional treatment can be
TH E RMAL A N A L Y S I S OF PWRS
1 70
--
--"' ...... ... _
x
(k +� ) ( dx
dX
)
2 dT + d T dX dx dx 2
dx
Fig. 2 . 1 7
Heat cond uction in a fuel pla te.
used. From a heat balance across a differential element (see Fig. have
d 2 T dx ) - k dT + qIIIdX = O (k + -dx dk ) ( dT 2 dx dx dx dx -+
2.17), we (2. 9 8)
where x is the distance from the fuel element centerline, q lll is the volumetric heat generation rate, and other symbols have their previous meanings. After eliminating the second-order term, we simplify to
( )
� k dT + q111 = 0 (2 .99 ) dx dx By integrating on the assumption that q lll is constant and by imposing the boundary condition that dT / dx is zero at the centerline, we obtain dT = - xq k(2.100) dx III
and finally
I k dT = �2 q'" x
_
(2.101)
Tc
where Tc is the center temperature. By substituting an appropriate expres sion for thermal conductivity, the temperature at any point can be deter mined. The assumption of negligible axial conduction can be justified for metallic fuel plates because of their very small thickness. For a U02 plate, the very low thermal conductivity of U02 provides the justification. The effect of axial conduction has been considered in detail by Fagan and Min gle,155 who provide expressions for surface heat flux. They indicate that for most power reactor systems, the minimum heat flux will be in error by <0.5% if axial conduction is neglected.
F U E L ELEMENTS
2 .3.1 .2
Effect of Flux Depression
In a uniform lattice, diffusion theory indicates thermal neutron flux
q'"
cosh(Kx)
(2.102)
where
q'" q�' =
q�'
cosh(Kx)
(2.1 03)
where is the volumetric heat generation rate at the centerline. In actual practice, flux depression probably will be computed by the escape proba bility method (see Sec. 1 .2.5) . Although a simple, closed-form solution is not obtained, numerical results generally can be fitted quite well by equa tions like Eq. (2. 102), provided K is appropriately chosen. If we proceed in the same manner as for a constant rate of heat gener ation, we obtain as the analog of Eq. (2.99)
:X ( k�:) + q�' COSh(KX) = 0
dT/ dx I k dT= :�
Integration twice with
=
(2.1 04)
0 at the centerline yields [cosh(Ka) - cosh(Kx)]
(2.105)
T5
q"
where a is the half thickness of the plate. We can also express our result in terms of surface heat flux per unit area by noting that
q" q�' =
hen ce,
K
sinh(Ka)
x - COSh(KX) ] I k dt q,, [ coSh(Ka) K sinh(Ka) =
(2. 106)
(2.1 07)
Ts
and -1] , fk dT= q,, [ COSh(Ka) K sinh(Ka)
Tc
Ts
where Tc is the center temperature.
(2. 108)
T H E RMAL A N A L YS I S OF PWR S
k
___ ...._ .
Fig. 2 . 1 8
2.3.2 2.3.2.1
+ ( dk ) d T dT dr r + dr dr d r dr r + dr
I
I
r + dr
Heat conduction in a fuel rod [from Nucl. Eng. Des . , 6, 301 (1967)] .
Heat Generation Within a Fuel Rod Constant Heat Generation Across a Rod
2. 1 8), S ( k + dkdr dr ) (r + dr) (dTdr + ddr22Tdr) -Skr dTdr + q"rS dr = 0 (2. 1 09)
From a heat balance across a differential annular ring (Fig.
we have
I
By rearranging and omitting higher order terms, we obtain
dT) +qll = O �( kr r dr dr
�
(2.110)
This can be integrated to yield
(2.111) kf dT= f [ 2qll ( r- r2; )] dr=4"qll [ (r2-r;)-2r; ( ln �r )] (2. 1 12) where Tc is the center temperature at central radius rc and Tr is the tem perature at radius r. -
T,.
When we are dealing with solid cylinders (no central hole), we can sim plify our last result to
qll R 2 kf dT=-_o 4 '
(2. 1 13)
F U E L E L EM E NTS
where Ro is outer radius and T" is the surface temperature. We can also express the volumetric heat generation in terms of the rate of heat generation per unit length of fuel rod q' as q' = 7rRJq'"
(2. 1 1 4)
the n (2. 1 15) For the approximation of a constant value of
k,
we obtain (2. 1 1 6)
From Eqs. (2.1 1 5) and (2. 1 1 6), we find that the temperature difference across a fuel pellet is independent of the radius of the fuel pellet. Specifying a given surface temperature and a given rate of power generation per unit length of fuel fixes the central fuel temperature. 2 . 3 . 2 .2
Concept of
f
k dT
Equation (2. 1 15) states that q' the rate of heat generation per unit length of a cylindrical fuel rod, is directly proportional to the thermal conductivity integral,
Thus, if values of the integral are available, the power level necessary to produce a given center temperature is readily specified. 1 s6 This is conven ient for the designer since it avoids the necessity of performing the inte gration of thermal conductivity values with respect to temperature. It is also a convenient way to report experimental data from in-pile irradiations. Under those conditions, power generation, center, and surface temperature can be known. Hence, f�c k dT is known, but pointwise values of are not available. Data are usualfy given in terms of f'6ck dT, where
k
(2. 1 1 7) Several curves of the integral as a function of its upper limit are shown in Fig . 2.19.
T H E RMAL A N A L Y S I S OF PWRS
u.. o o o o lO N
o 1 00 �----��-----+--�--+-- �
90 80 E
u
3' ;;
70
..
�
u ....
�o
60
40 1-----=:!itr-*---:.II"-4:-----+--+--t---1
� ���+----+---+- �--_t_---_1
-�---�----+---+_1 0 ��- ---r_--__1
1 000
Fig. 2 . 1 9
�oo
3000 Tem perature,
4000
5000
0F
Summary of Jk dT for U02. Notation: 1. Weidenbaum, GEAP-3771-8, General Electric Company (1963). 2. Duncan, CYNA-142, Westinghouse Electric Corporation (1962). 3. Reference 40. 4. Robertson et aT., J. Nucl. Mater., 7, 3, 225 (1 962). [From McNelly, "Liquid Metal Fast Breeder Reactor Design Study," GEAP-4418, General Electric Company (1963).
6000
F U EL E L EM E N TS
While the concept is of particular value for U02 fuel rods, it is also useful in describing U02 fuel plate behavior. As seen from Eq. (2.101 ), a knowl edge of the integral and plate thickness is sufficient to establish the volu metric power generation required for a given temperature. Indeed, it is possible to relate dT to the heat output for any geometric shape. Thus, for constant heat generation
Jk
f k dT = Cq '
(2.1 1 8)
T"
where C depends on fuel geometry. Specification of the maximum allow able value of the integral fixes the power capability of the fuel in any given geometry. As previously noted, a common design criterion is the avoidance of center melting. Since this is an observable point, a series of tests at var ying power levels can be used to determine the dT at which this first occurs. Observations are not dependent on a precise knowledge of fuel conductivity or temperature distribution within the fuel. Values, deter mined by several investigators, for the dT to produce melting are shown in the upper right of Fig. 2.19. More recently, Lyons et aI. I 57 evaluated this integral in a series of short-time, in-pile tests. From metallographic exami nations they concluded that for 95% dense, sintered U02 pellets, the aver age value of the conductivity integral from O°C to melting was 93.5 W cm. Statistically determined 95% confidence limits were ± cm. Direct measurement of the integral based on a gas-bulb sensor yielded a value of �90 W cm. Vibratory compacted fuel was found to have a significantly lower integral even when the density correction was made. Expressions for the thermal conductivity integral for sintered pellets have been provided by Lyons et aP 57 and MacDonald and Thompson,53 among others. For 95 % dense U02 with temperatures between 0 and 1650°C, MacDonald and Thompson suggest
Jk
Jk
3.5 W /
/
f k dT = 40.4 ln(464 + T) + 0.027366 exp(2.14 X 1 0 - 3T) - 248.02
/
T
(2.1 19)
o
For temperatures between 1 650°C and melting, T
f k dT = 0.02(T - 1 650) + 0.027366 exp(2.14 X 1 0 - 3T) - 0.93477
1650
Jk
/
(2. 120)
In these equations, T is in °C and dT is in W cm. These equations yield 95.4 W cm as the conductivity integral from O°C to melting. For temperatures above 773 accurate values of the thermal conductiv ity integral can be obtained by integrating the analytic equation of Harding
/
K,
T H E RMAL A N A L Y S I S OF PWRS
1 76
and Martin42 [Eq. (2. 10)]. With the polaron contribution considered as neg ligible below 773 K, this integration yields
f k dT = 4.61 9 x I03 In[O.1 830 + 1 .0566 X I0 - 3T ] T
773
- 4.7 1 5 X 1 09 exp ( - 1 6361 / T)
4 ( - 1 ) i (4) ! (1 /T)(4 i) ! ( - 16361)( i + 1 )
� i O (4
K
-
_
(2.121)
for 100% dense U02 . In Eq. (2.121), T is in and k is in [W / (m K)] . Harding and Martin42 suggest that, when the integral is applied to less than 1 00% dense U02, the integral be multiplied by (1 - 2.5a) where a is the void (porosity) fraction. The integral of Eq. (2.121) may be used to eval uate f'6 for temperatures above 773 K with Eq. (2.1 1 9) providing values of the integral below 773 K. The effect of long-time irradiation on the conductivity integral is uncer tain. While most earlier observers concluded there is significant reduction due to irradiation, some data dispute this.47, 158 Stora ' s previously cited ex planation45 that most apparent effects are simply due to thermal cracking and, thus, depend on the available gap, seems most reasonable. However, as noted in Sec. 2.1 .1 .3, there remains some irradiation effect after the effect of cracking is removed. The thermal conductivity correction factor of Eq. (2.12) can be used to correct the integral for the residual irradiation effect. Soulhier159 used the conductivity integral for correlation of fission gas release data. His results are presented as
f
f
k dT (max) versus F k dT
400°C
Ts
where F is the fractional gas release, Tc is the center temperature, and Ts is surface temperature. The integral froo0c k dT is used to normalize all experiments performed at a given central fuel temperature to the same surface temperature of 400°C. The product F fEk dT is used to normalize the fission gas released based on the volumetric average heating rate. 2.3.2.3
Rod with Flux Depression
We previously showed that diffusion theory indicates thermal flux distri bution across a cylindrical fuel rod in a uniform lattice given by (2. 122)
F U E L E L E M E NTS
where A =
1 0
K
= =
constant zero-order modified Bessel function of the first kind the reciprocal of the thermal neutron diffusion length in the fuel = VLa /D 1 .
K
When the escape probability method is used, which is more usually done, should be regarded as a constant selected to yield a closed-form approximation of the results. However, the experimental work of Gibson and Anno 160 indicates that measured values of differ by only about from that obtained by simple diffusion theory. Since flux depression effects are small at PWR enrichments, the error in would cause only a very small error in the fuel temperature. Since is proportional to
K
q'" q�'
K
10%
q'"
Now let be the value of at the center of the pellet. Since previous expression becomes
The average value of
q'"
(2.1 23) 10(0) 1, (2. 1 24) =
over the entire rod cross section is given by a
f q"'(27Tr) dr
Ii''' = 0
a
our
� f qlllr dr a2 f 27Tr dr
(2. 1 25)
a
0
o
where is the outer radius of the pellet. By substituting our expression for local flux in Eq.
[II (Ka) q 2q�(Ka) (2. 1 10)] [ll (Kr)] dT = 0 q�'r(Kr) dr
(2. 1 25),
we obtain
]
_ ",
(2.1 26)
Substitution of the heat flux expression in our equation for the temperature gradient at any point [Eq. and integrating yield +k
On separating the variables and integrating from
r
(2.1 27) to a, we obtain
T H ERMAL A NA L Y S I S OF PWR S
1 78
(2. 1 28) If we rewrite the above in terms of the average heat generation rate, we get
(2. 1 29) (2.1 29)
At the center of the pellet, becomes
r = 0 T = Tc and
Under these conditions, Eq.
(2. 1 30) If we let and replace q lll in terms of rod [Eq. we obtain
(2. 1 14)],
q'
f= 2[Ka[Io(Ka)I} (Ka)]-1]
(2. 1 31)
the rate of heat generation per unit length of
(2. 1 32) f,(2.115) (2. 1flux 32). depression factor, f 95% K 3 -} 2.5 6%
The term which is the becomes clear when we com pare Eqs. and We see the thermal power associated with the given fuel surface and center temperature is greater for nonuniform than for uniform heat generation; the ratio of the two heat ratings is f. Approximate values of can be obtained from diffusion theory if values of are available. For a U02 of theoretical density, varies from to 235 U. Figure cm for enrichment values of to shows as a function of enrichment and fuel diameter for pellets of dense U02 . For the enrichments used in PWRs the value of f is not far from unity. However, this effect should not be ignored since it can mean an increase of several percentages in reactor power output. The fact that is close to unity can be used to justify the simple approximation that thermal flux inside the fuel follows a parabolic distribution. If this is done, values of F (the ratio of surface-to-average flux in the rod) obtained from escape probability theory can be used directly to compute without assigning a value to Conse-
K2.20 95%
f2
f
f
K.
1 79
F U E L E L E M E NTS
1 .0
..
...."..."'""""'-----,
0.95
:-'�l 1
0.90 Fue l Rodius,
cm
0.85
.=
0- 0 '" '"
::: ...... 0.80
L--J
0.75
0.70
4
o
Fig. 2.20
Aux
6
10
12
Enrichment, wt% ZlSU in Totol U
14
16
20
depression factors for V02 fuel rods of 95% theoretical density [from Ref. 156].
quently, for a parabolic flux distribution
f=(3-F)/2
2.3.2.4
18
(2. 1 33)
Heat Generation with a Power Tilt or Eccentric Pellet
1.3.2,
In Sec. we indicated that flux distribution across a cluster of rods in a pressure tube of 020 or a graphite-moderated reactor is approximated by the function When a single rod is considered (Fig. this ap pears as a flux tilt. A reasonable representation of the distribution has been found to be161
10(r).
2.21),
q"(r,e)= q"'(1 +E' r/a
cose)
(2. 1 34)
where E ' represents half the difference between the minimum and the max imum heat generation rate divided by q III , the average volumetric heat gen era tion ra te
(2.1 35)
Since we now consider both radial and azimuthal variation in temper ature, we must use the general form of the steady-state conduction equation
(2. 1 36)
T H E RMAL A N A L YS I S OF PWRS
1 80
Fig. 2 . 2 1
Fuel rod with flux tilt.
In cylindrical coordinates with the present heat flux distribution, this becomes
a2T2 +-"iJ"' ( 1 + £, - ) =0 -1r -dra ( r-aTar ) + -r12 ae k a
(2. 1 37)
i dT' a2T' r ar ( r ar ) + �r 2 ( ae2 ) =0
(2. 1 38)
cose
in the pellet region, if we assume a constant value for k.* Since we have no heat generation in the cladding region we obtain
!
T'
where represents the cladding temperature. At the boundary of the pellet and cladding, there is normally a gap to which we assign a conductance hg . Since heat transferred across the pellet, gap, and cladding must be equal to we have the boundary conditions at r = a : hg
k
k'
r = a, [T(a,e) -T' (a,e)] - k (��) I , � k, aT'dr 1 _kaTar I =
r=a
r=
(2. 1 39) (2. 1 40) r=b (2. 1 41)
where and are the thermal conductivity of the fuel and cladding, respectively. If the heat transfer coefficient to the coolant is high, at we say
T' (b,e) Tbu1k =
wate r
2 ""Andrews and Dixmier l 6 provide a solution for the case of varying k.
1 81
F U E L E L E M E NTS
(2. 1 37) (2. 1 38)0, 4k
Also note that at r = T is not infinite. With these boundary conditions, Eqs. and can be integrated to yield - lIIa 2 b - III In- + \)J(r) cos8 + Tb T = L (a 2 - r 2 ) + - aq lll + hg k' a and
2
1
q2
(2. 1 42)
-lIIa 2 b T' = � In- + \)J' (r) cos8 + Tb r k
2
(2.1 37)
(2. 1 42)
We must now evaluate \)J(r) and \)J' (r) by substituting Eq. and solving the resulting differential equation to get a \)J(r) = Ar - - r 3
8
where a = E'q lll /ak. Similarly, if we substitute Eq. can solve to obtain
CO r/b2) (2. 1 39) (2. 1 40),
\)J' (r) = Using these results in Eqs. A and C :
(2. 1 43)
into Eq.
/r -
and
(2. 1 43) into Eq.
(2. 1 44) (2. 1 38) (2. 1 45)
we
we can solve for constants
(2. 146) and
C 4 =
aa4
!5:.-r k'
(
b2 / (b2 + a2)
k b2 - a 2 1+ +- � hg a k' C/ + a k
)J
(2.147)
If desired, the maximum fuel temperature at any radius can now be deter mined. This occurs at 8 = O.
1 2g
(
E , q lll 1 + 3� q lll q llla2 b a 2r - r� Tmax = - (a 2 - r'; ) + - aq lll + - In- + Tb + -k k' a 8ak 1 + � m h
4
where
2
---
) (2. 148)
(2. 1 49)
k k b2 - a2 �=-+h cf1 k' b2 + a2 , and rm , the radial location of the maximum fuel temperature, is given by -
T H E RMAL A NA L Y S I S OF PWRS
1 82
m �
;:' ([ 1 + 3/4(£,)2 (\:3��) r -1 }
We can also determine the maximum heat flux [Btu / (h ft2)] that also occurs at e
= 0,
(2. 1 50)
Hence,
abE' -- =1 + ---(1 + �)(b2 + a2)-
(2. 1 51)
q':nax
For
b = a + dr'
q ';
a � dr' (2.1 52) = 1 + [ k (E'1 dr' ) ] 2 1 + -a h-g +-k' E = 0.2, k=1. 6 a=OA h =1000, dr' =0. 0 13, g � 1.09. where
the last equation simplifies to
q ':nax q ':nean
k'=12
________
An example of what is obtained in a typical case is useful: Assume Btu/h ft OF, Btu /h ft OF, in., then q max / q mean is Analytic approaches can also be used for solution of the eccentric pellet problem with a uniform pellet conductivity. The nonhomogeneous differ ential equation can be transferred into Laplace' s equation by a change in variables. 1 63 The resulting partial differential equation may then be solved by the separation of variables technique. However numerical approaches using finite differences are more usually used. Sdouz and Dagbjartson164 have shown that, with an open pellet to cladding gap, an eccentric pellet that touches the cladding can lead to noticeable circumferential variation in the outer fuel temperature [about when operating at kW / ft] but almost no change in cladding temperature. The significant circumfer ential variations in gap conductance assumed here would only arise at the very beginning of fuel life. As noted earlier, significant fuel relocation oc curs after the first rise to power and after hr the gap is largely closed. Since cladding temperatures are unaffected and fuel relocation rapidly re moves the azimuthal fuel temperature variations, pellet eccentricity does not appear to pose a real problem.
200 K
13
500
2 . 3 .3
Annular Fuel Elements
a b, ] 4> =C ,
For a hollow cylinder of outer radius and inner radius calculations give the flux distribution as
diffusion theory
(2.1 53)
1 83
F U E L E L E M EN TS
where
Ko == Kl = C'
zero-order modified Bessel function of second kind first-order modified Bessel function of second kind at r = a. constant evaluated so that
For heat removal at the outer surface only, we write (2.154)
Ts
When heat is removed at both surfaces, the situation is complicated if dif ferent boundary conditions hold at the two surfaces. When is the same at both surfaces, we get T,
f k dT= Ts
}.. = [1(Kb)/K1 (Kb) RORO Ro (�) [11 (KRO) - AKI (KRO)] = � [10(KtI) -Io(Kb) � [Ko(KtI) -Ko(Kb)
where and perature occurs. We obtain In
2 .4
(2.1 55)
is the radius at which maximum fuel tem from I+
I
(2.156)
REAL I STIC EVA L U AT I O N OF STEADY-STATE TEMPERATU RES
Fuel element temperature calculations of the previous section did not con sider the slowly varying changes in fuel element properties that take place during reactor exposure. After the initial rise to power, cracking of the fuel pellet changes fuel conductivity and gap conductance. Gap thickness de creases with time as the Zircaloy cladding creeps down toward the fuel pellet. In most long irradiations, fuel swelling eventually brings the pellet into contact with the cladding and further increases gap conductance. In addition, at high heat ratings, a central void gradually forms. Because of these gradual changes, fuel element temperatures are a function of the
T H E RMAL A NA L Y S I S OF PWRS
1 84
power-time history of the fuel rod. Realistic fuel temperature evaluations must include all of these time-varying effects. In view of the complexity of the phenomena involved, satisfactory closed-form analytical solutions are not available*; numerical procedures are required. The fuel pellet is usually divided into a series of concentric annular rings. Using an assumed temperature distribution (or that from a previous time period), the response of each of these rings to the combined effects of thermal expansion, swelling, void migration, creep, and hot press ing during some short time period is determined. The interaction between pellet and cladding is then determined to establish gap conductance. Tem perature distribution is then redetermined. If the new distribution is close to that originally chosen, the calculation proceeds to the next time interval. If agreement is poor, behavior of the fuel rings is redetermined and a re vised temperature distribution is obtained. The process continues at a given time step until convergence is obtained. 2 .4 . 1
Rigid Pel let, No Fuel-Pellet Friction
The simplest situation arises when the center temperature is below �1450°C. Under these conditions, fuel restructuring does not occur, no cen tral void is formed, and fuel creep and hot pressing are minimal. Thus, the change in fuel dimensions is often taken as the sum of that due to thermal expansion and swelling. An initial temperature distribution is assumed; thermal expansion and swelling of each ring is evaluated. Summation of the ring dimensions provides the new fuel pellet diameter. This is com pared to the cladding diameter and the gap width is then determined. If a sufficiently large number of rings are chosen, properties are assumed constant across each annular ring. In one procedure}66 it is assumed that heat flux q�v and thermal conductivity kay can be evaluated at the center of the annular ring and that the equation for the temperature drop across a slab can be used. Then the temperature drop across an annular ring, 6.Tr , is given by
(2.157) where 6.r is the thickness of the ring. When fewer rings are chosen, more accurate results are obtained by modifying Eq. (2.1 12) to
f o
dT= f k dT + Tr2
r1
k
0
{�' [(ri - rf) - 2rf(lnr2 /rl)1 }
(2.158)
*Lamkin and Brehm 1 £>5 provide an exact closed-form solution for fuel temperatures and stresses, providing simplified creep and property formula tion are accepted.
F U E L E L E M E NTS
1 85
where Trl and Tr., equal temperatures at the inner and outer radius of the ring. If the temperature of r2(Tr) is known, f,[;r2k dT is known for a given porosity. Then fb'] k dT can be obtained and Trl is known. Both of the above schemes are iterative. A surface temperature of the pellet must be assumed. The calculation then proceeds inward. The final temperature distribution is then used to compute a revised estimate of fuel dimensions and gap conductance, hg . With a new hg , a revised pellet surface temperature can be computed. At low burnup, irradiation growth of the cladding is not significant. However this growth should be considered for high-burnup fuel. Thus, cladding strain is the sum of thermal expansion, irradia tion growth, and elastic and creep strain. Circumferential cladding strain, fB ' is given by E" =
(�)
[ a. - "I(a , + a, )] + "cU' T ) + t, + ti,
(2. 159)
where IT r ,IT s,ITz
E
Uc fc
fir
= = = = =
-y =
dT
radial, tangential, and axial stresses, respectively Young ' s modulus coefficient of thermal expansion creep strain strain due to irradiation growth Poisson' s ratio increase in cladding temperature.
b
Given a cladding with outer radius and inner radius a, and the absence of pellet-clad contact, the stresses are obtained from (2.1 60) (2.161) (2.162) where
Pi 'Po
inner and outer pressures, respectively E / [2(1 + -y)] r = radius at a given point A = 2G/(1 - 2-y) E = Young' s modulus.
G
=
=
T H E RMAL A NA L Y S I S OF PWRS
1 86
Since a and b do not differ greatly, stresses at an average value of r can be used to estimate an approximate value of E,. , the radial cladding strain. The value of Po is determined by external system pressure. When the fuel pellet is not in contact with the cladding, Pi is the pressure of the fission and fill gases inside the fuel rod and can be calculated via Eq. (2.57). Evaluation of the quantity of total fission gas release requires an esti mation of gas released from each radial node at a number of axial eleva tions. Once convergence on surface temperature at a given axial elevation is obtained, the process must be repeated at a number of other axial ele vations. Fission gas release must then be determined. If significant changes in internal pressure or gas thermal conductivity are found, the process must then be repeated at all elevations. Cladding creep will not occur when the cladding is in a hydrostatic stress state; i.e., ae = a r = az . According to the Prandtl-Reuss assumption, creep is determined by deviatoric stress, s . In the circumferential direction,
(2.163) where a lll = mean stress = (1 /3)(a,. + az + ae). To relate the multiaxial situa tion to the uniaxial condition in which strain rates are normally measured, an equivalent stress, aeq' is defined as
(2.164) The equivalent stress is proportional to the total shear stress, which gives a measurement of total plastic creep deformation in a polycrystalline ma terial. The value of aeq is substituted in an appropriate uniaxial creep equa tion, e.g., Eq. (2. 9 2), to determine an equivalent creep rate £�q . Creep rates in the principal axes are then obtained from
(�)
. r / sr - Ece / s e Ezc / Sz - (E. ec q / ae ) q 2
Ec
_
.
_ -
.
_
(2.165) (2.166)
Total creep strain during the time interval can then be estimated from the product of £� and �t. The procedure is satisfactory as long as small creep strains are obtained. If large creep strains are encountered, the time step must be reduced. If the total cladding strain computed from Eq. (2.15 9 ), using inner pres sure as gas pressure, leads to the conclusion that cladding diameter is equal to or less than the uncracked diameter of the pellet, then the fuel and the cladding are in contact. The fuel will elastically strain the cladding so the pellet and cladding are just in contact. The elastic expansion, �Di ' of the cladding is
F U E L E L E M E NTS
1 87
(2.1 67) where Di Q is the internal diameter of cladding at BOL conditions and Dp is the uncracked diameter of hot pellet, including swelling effects. Internal pressure, Pi ' required to obtain this expansion, can be computed from Eq. (2.65). As previously noted, contact pressure equals Pi less internal gas pressure. The contact pressure is then used to evaluate gap conductance. If one makes the assumption that interaction with the pellet stack does not restrain the vertical movement of the cladding, then Eqs. (2. 1 60) to (2.162) can be used to obtain the stresses from the external pressure by using an average value for r. Again an iterative procedure is required to obtain agreement between the assumed and calculated cladding and fuel movement. In estimating the fission gas pressure, appropriate temperatures must be assigned to the various gas volumes. The gas in the fuel pores and dish volumes can be assumed to be at the average fuel temperature while the gas in the gap is obviously at the average gap temperature. The gas in the upper plenum is sometimes taken to be at the temperature of the exit cool ant. However, this assumption underestimates the plenum gas temperature because it does not allow for (1) the heat supplied by the gamma heating, Qs' of the hold-down spring located in the plenum and (2) transfer of heat from the uppermost pellet. This heat must be transferred to the cladding by natural convection. Since natural convection heat transfer resistances are high in comparison with the thermal resistance of the cladding and coolant film, we may write (2.168) where
rp
=
he
=
hp
TC lTf,Tp/
=
=
pellet radius natural convection heat transfer coefficient between top pellet and plenum gas natural convection heat transfer coefficient between cladding and plenum gas temperature of coolant, fuel, and plenum gas, respectively.
Plenum temperatures calculated from Eq. (2.1 68) are noticeably above the coolant temperature. Since most of the fission gas is in the plenum, this temperature has a significant effect on the calculated gas pressure. 2.4 .2
Rigid Pellet, Fuel-Pellet Friction Considered
Und er most circumstances, the assumption of no friction between the fuel and cladding is unwarranted . Since the fuel cracks shortly after the first rise to power, there is resistance to relative motion between fuel and clad-
T H E RMAL A NALYSIS OF PWR S
1 88
ding even when there is a nominal open gap. The simplest assumption that can be made is that the friction between fuel and cladding always prevents significant axial strain of the cladding. If the axial strain is largely elimi nated, then the axial stress is relatively small. Further, the radial stress is relatively small in comparison with the circumferential stress. If both Uz and Ur are considered negligible, then the equivalent stress, ueq, of Eq. reduces to Ue . The circumferential creep rate from Eq. then reduces to the creep rate from the appropriate uniaxial creep rate equation with Ue as the uniaxial stress. The average circumferential stress to be used may be computed from thin-wall theory [Eq. The assumption of a negligible axial stress can be improved on even when thin-wall theory is used. If the axial strain of cladding is set equal to the axial strain of the fuel obtained from thermal expansion and net swell ing, Ez for the cladding is fixed. The radial displacement at the inside diameter of the cladding, ilr;, is determined by the radial movement of the outer radius of the pellet. If we follow the treatment of Berna et al., 1 20 the displacement of the cladding, r, is related to the circumferential and radial strain by
(2. 1 64)
(2. 1 66)
(2.78)].
(2. 1 69)
2
C Er ilr·1 = rEo. - u
where C is the cladding thickness and r is the average radius of the clad ding. By taking the average radial stress as negligible, the strains are then related to the stresses by
(2.1 70) (2. 1 71) (2. 1 72) where EPe, Ef�, E� are the permanent plastic and creep strains in the S, Z, and r directions; E�� E� E �h are the thermal strains in the S, Z, and r directions; and 'Y is Poisson' s ratio. By substitution of Eqs. and in Eq. we obtain il r
(2. 1 69) (2. 1 72) (2.170), ) ( ) [ ( 2 ) ( 2r ) ] (2. 1 73) (173) (171)
i + C E P + E t.h - E P + E th = � r I ij ij E r
2(
Simultaneous solution of Eqs.
l + 'Y C uij + 'V C - l Uz I r
and
yields
1 89
F U E L E L E M E NTS
where
An
�
1+
��,
A' 2
�-Y(�r )
-1 -
A2 1 = - 'Y , A22 = 1 ,
B2 E [Ez -(E� -E;h)] =
The stresses thus obtained can then be used in Eq. (2. 164) to determine the equivalent stress and from this stress the creep is determined. The internal is determined from pressure,
Pi '
CUtl - roPo Pi = --- ri
(2.174)
where the subscript "a" indicates the values at the outer radius. As previ ously indicated the contact pressure is the difference between and the gas pressure. An iterative process is required to obtain an agreement be tween the stresses and the cladding motion just as for the open gap case. Berna assume that the cladding is free to move relative to the fuel when the gap is open. However, if the high-power pellets in the center of the rod are in firm contact with the cladding, those pellets in the lower portion of the rod are trapped. The FRAPCON and FRAP-T programs 1 67 assume that the axial expansion of the fuel will still be imparted to the cladding. The total change in the cladding length, £1 L , is computed from the sum of the pellet length changes. This £1 L must equal the sum of the length changes of the individual axial segments. That is,
Pi
et aU 2D
n
� L = .2: J
=
1
[(Ez) j - (E�) j] (£1z) j
(2.1 75)
where i' j are the axial cladding strains on the j' th segment at the beginning and end of the current time step and ( £1z) j is the length of axial segment j. The axial stress is taken as constant along the trapped pellet stack. By expressing the axial strains for the individual segments in terms of the stresses we obtain
(E�) (Ez)
Uz
(A
( D)
( P)
(
Ue) j + E z E z j Ezth)j ] } (2.1 76) ( i { ['Y z = [ j�l� �] ) £1 L + � j Z j�l u Ej (� Z ) j where the subscript j again indicates the j' th axial segment. Since the pellets are not in contact with the stack, Ue may be obtained from Eq. (2. 1 74) using the gas pressure Pi and the internal and external radii appropriate to the given node. The elastic axial cladding strains are related to via the value IT
of E for the particular segment.
j
_
_
IT:
T H E RMAL A NA L Y S I S OF PWR S
1 90
An alternative way of dealing with the effect of pellet interference on cladding motion is through the assignment of a frictional force between pellet and cladding. 1 68 In the upper region of the rod, the axial force, Fj, on the cladding at an elevation j is then given by (2.1 77) and
F.. = f I
.
J-
1
+
Ff"
(2. 1 78)
where
Fl
=
force due to plenum spring Pg = internal gas pressure rp = pellet radius = frictional force.
F';
The maximum frictional force, �:
(F';- )max,
depends on the contact pressure, (2.1 79)
where � is the coefficient of friction and K is the frictional force per unit area at zero contact pressure (often set at zero). If the pellets are in contact, the axial stress, uZ' obtained assuming the cladding strain and pellet strain are equal, is first determined. The value of (Jz so determined is used to compute an alternative Ff from (2.1 80)
and Eq. (2.178). If Fi so determined exceeds (Fi)max, then pellet slippage is possible and (Fi)max is used to reevaluate Uz. Assignment of a maximum fric tional force depending on contact pressure means that when contact is first established with a low contact pressure there is not an abrupt change in Uz. 2 .4.3
Behavior Considering Pellet Plasticity
At the high linear power at which fast reactors operate, columnar grain growth with central void formation is a significant factor affecting fuel rod behavior. However, if we use Nichol' s calculations of the columnar grain radius presented in Fig. (2. 1 1 ), we find that a temperature of over 1 800 K is required to produce any columnar grains after very prolonged exposure (30,000 hr). At PWR coolant conditions, this would correspond to a linear power rating of about 15 kW / ft. More recent experimental evidence8? in dicates that temperatures in excess of 2100 K are needed for significant columnar grain growth. This would require operation at about over 16 kW / ft. Current (1995) PWRs do not operate at peak powers this high (usual
F U E L E L E M E NTS
1 91
range 1 2 to 1 5 kW 1ft) and hence columnar grain growth and central void formation are not observed.87 Although void formation is not a problem in PWR fuel, central temper atures in the hotter fuel rods are sufficiently high so that the central fuel region exhibits some plasticity. A complete model of fuel behavior would need to take this plasticity into consideration. To do so, a stress and strain analysis of the fuel pellet is required. The stresses so calculated are used, with an iterative process, to obtain the plastic and elastic strains within the fuel. The computation must include relationships expressing these strains as a function of applied stress. The FRACAS167 deformable pellet model calculates the pellet stresses and strains by an approach somewhat similar to that for cladding strain computations given by Eqs. (2.1 69) through (2.1 73). At each axial segment the pellet is divided into a series of concentric annular rings. A generalized plane strain deformation is assumed so that Ez is a constant for all radii. The value of this strain is subsequently determined from the requirement for axial force equilibrium. The governing equations of equilibrium and compatibility for each ring in the absence of any displacement discontinuities are: (2. 18 1 ) dEe Eo - E r -+ =0 dr r --
(2. 1 82)
The elastic-plastic stress-strain relations, which are similar to those of Eqs. (2. 1 70) to (2. 1 72), are (2.1 83) 1 E0 = - [a e - "Il(a r + az )] + Eth e + EPe + dEPe I E
(2. 1 84) (2.1 85)
Substitution of Eqs. (2. 1 83) and (2.1 84) into Eq. (2.1 82) yields
(2.1 86)
T H E RMAL A N A L Y S I S OF PWRS
1 92
£z
(2. 1 81), (2.1 86), (2. 1 85)
with constant relate the stresses as and Equations they vary across the pellet. The stresses are evaluated at the midpoints of each ring after writing the foregoing equations in finite difference form. A set of recursion relations,
{::L [LU)] {::} �
+
(2.1 87) +1
{MU)}
to is then obtained . This matrix equation relates the stresses at node i depend on the material prop and those at node The matrices erties, geometry, and plastic strains only. a relation between the stresses By successive application of Eq. at any node and the stresses at node the node at the inside of the cylinder, can be obtained . This relation takes the form 1
i.
[L(i)] {MU)} (2.I, 1 87),
{::Uz } + 1 = [A U)] {U::z} + {B(i)}
[A U)][A U)] {BU)}{B U)} 1
The matrices recursion, the result that
"1" N
(2.1 88) [L(O] {M[i]}.
and can be determined from By across the cylinder wall can be obtained, with
and and
{::U=} =[A (N+l)] {U::z} +{B(N+l)} N
(2.1 89)
1
where is the index of the ring at the outside surface of the cylinder and - Po, where Po is indicates the inner ring. At the outside surface, the gap pressure if the gap is open or the contact pressure if the gap is closed . The following condition can then be obtained: - Po
Ur=
= A (N - 1)ur(1) + A liN -1)uu(1) + A 1 3(N -l)uz(1) + Bl (N -1)(2. 1 90) U L Uz (j) dA(j) =Fz (2. 1 91) 11
At the innermost pellet ring, ur is set equal to The condition of axial equilibrium
1.1 .
is used to determine the stress at the midpoint of the inner ring. By use of the recursion matrices this becomes
(2. 1 92)
F U E L E L EM E NTS
1 93
After determination of the total axial force by essentially the same approach as shown by Eqs. (2.1 77) and (2. 1 78), the stresses at the inner node are ob tained by simultaneous solution of Eqs. (2.1 92) and (2. 1 9 1 ) with the boundary condition that (J"rO ) =
{M(i)}
T< T�
and
(2.1 93)
and
(2.1 94)
is the stress along the principal axis (r,8, z), T� is the brittle where
et
T ?T:
The presence of cracks sharply reduce Young's modulus for the fuel. Some calculational schemes simply set Young' s modulus to an arbitrarily low value in the cracked region. However, other approaches attempt to correlate E. Sakagami al. 1 68 define E' as an effective value for Young' s modulus and "I ' as an effective value for Poisson' s ratio. They conclude
et
( ) (--) 2
E' - 3
n-l
E
1 - "1
(2. 1 95)
(2.196)
n
where is the number of principal directions in which there are cracks [determined from Eqs. (2. 1 93) and (2. 1 94)] . Willford, 1 70 however, concludes that E' is a function of stress and represents E' by (2. 1 97)
where s is a constant,
T H E RMAL A N A L Y S I S OF PWRS
1 94
6 1 0 �____________________________________________�
10
�
6
S O L I D U 0 2 AT
R O O M T E M P E R AT U R E ,......
0
:E p..
10
4
'-"
CI)
�
....l
� Cl 0
::E
1 0
3
CI)
UJ Z ::> 0 >t.Ll >
1 0
E-<
U
t.Ll � � t.Ll
1 0
1 0
2
-
-
I
D 0
10
-2
1 0
10
0
10
I
St u d y
1
St u d y 2 Study 3 E z: E D
I
1 0
(j' 2
1 0
3
C O M P R E SS I V E ST R E S S ( M Pa)
Fig. 2 . 2 2
The effective Young's modulus for cracked U02 fuel pellet columns [from Ref. 170].
plastic-elastic un cracked ring and an outer elastic but cracked region1 68 (see Fig. The contact pressure, P is determined by means of so-called "compliance" terms. The compliance represents the elastic strain produced by one unit of pressure. Hence, Per the contact pressure, is calculated by l 1 2
2.23).
(2. 1 98) where .1 D; = required elastic expansion of cladding to accommodate expanded pellet diameter
1 95
F U E L E L E M E N TS
I n t e r m e d i ate
E l ast i c - P l as t i c R e g i o n
P l as t i c R e g i o n
O u te r
C rac ke d Reg i o n
Fig. 2 .23
D= We' WFF =
Three-region pellet model.
hot restrained fuel diameter compliance terms for fuel and cladding, respectively.
The compliance term for thin-wall cladding is given by 1 1 2
Di, Do (2)
D;+Dn W = 2Ec(D o -Dc) c
-----
(2. 1 99) Ec 2.23).
where are the inner and outer diameter, respectively, and is Young' s modulus for the cladding. The fuel compliance term is the sum of (1) the compliance of the intermediate cylinder for deflection of its outer radius and the compliance for the outer cracked region (see Fig. If the fraction of the outer pellet surface with an open crack width is small, then l 1 2
(2.200) wh ere
D (DF)eF ==
hot outer diameter of the fuel pellet inner diameter of cracked pellet region (outer diameter of elastic region)
T H E RMAL A N A L Y S I S OF PWRS
1 96
( D F)p
=
outer diameter of plastic pellet region (inner diameter of elastic region) Young' s modulus for fuel in outer cracked pellet region Young ' s modulus for intermediate ring of elastic fuel Poisson ' s ratio for fuel in intermediate elastic ring = radial pressure at (DF)F radial pressure at ( DF) c-
Ep Ec "Ii Pi Po =
The first term on the right-hand side of Eq. (2.200) represents the compli ance of the cracked pellet region, while the second term represents the compliance of the intermediate ring. Although the intermediate ring has enough plasticity to close the crack, elastic behavior is assumed in the cracked region. Therefore the pressure at diameter DFe is given by
Po
Po= (�:J p,
-
(2.201)
The stress intensity, which in this case is l
Pi
et al.} 68
2 .5 2.5.1
F U E L P E RFORMANCE Typical Reactor Operating Conditions
Typical average linear power and average reactor power densities are listed in Table 2.VII fOF both conventional PWRs and CANDU reactors. Note that the newer 1 7 X 1 7 PWR assemblies and the 37-pin CANDU have somewhat
1 97
F U E L E L EM E N TS
TA B L E 2.VI ! Typical Fuel Operating Conditions
Assembly design Rod diameter (mm) Average linear power (kW 1m) Average power density* (kW 11)
Conventional
PWR
1 5 x 15 10.7 23.1 1 06
1 7 x 17 9.5 1 7.8 1 05
CANOL 2 8 pin 1 5.2 26.5 85
::.7
pin
13.1
:'5.7 ,19
"'Based on total fuel element and coolant volume.
lo wer linear power ratings than the earlier 15 X 15 and 28-pin desi gns. The lin ear power ratings are based on the total reactor power and do not allow for the small amount of power generated outside the fuel (abou t 2.6% of total power in a typical conventional PWR). At the usual operating conditions of a conventional PWR, the maximum cladding surface temperature is about 347°C. Measurement'j3 of fuel tem peratures of typical helium-filled rods with 96% dense pellets at usual cool ant conditions indicate centerline temperatures of 1200°C at 33 kW /m (�10 kW/ ft) and 1 760°C at 49 kW /m (�15 kW/ ft). The measuremen ts were taken after sufficient burnup for pellet cracking but before hard cladding to pellet contact was established. With a 2.82 value of Q' as used in several conventional designs, the average linear power rating for the 1 7 X 1 7 design shown in Tab le 2.VII converts to a maximum linear power of 50.2 kW 1m (1 5.3 kW 1ft) . The ex perimental observations reported above then indicate a maximum center line fuel temperature of about 1800°C. The maximum fuel temperature at the first rise to power (before pellet cracking) would be expected to be about 200°C higher. With fuel temperatures that exceed 1 800°C for only a brief period, one would not expect columnar grain growth but there would be a small equiaxed grain region near the center of the hot pellet. At the av erage linear power rating, no such grain growth would be observed. If the temperature profile is taken as approximately parabolic, the vol umetric average fuel temperature of the hot pellet is approximately 1055°C. The fractional fission gas release from the hot pellet would be less than 15% at moderate burnups (see Sec. 2.2.4.3). Average fractional fissio n gas re leases for the entire rod, assuming the normal operating power is not ex ceeded, would be expected to be no more than 5 % . However, conservative design calculations are usually based on assuming operation at about 1 1 0% of nominal power and these would result in somewhat greater ca l culated relea ses.
F
2 . 5 .2
Fuel Behavior
The objective of all steady-state fuel performance analysis is to establish fuel element designs and performance specifications that lead to a low prob-
1 98
TH E RMAL A NA L Y S I S OF PWRS
ability of fuel element failure. A fuel failure may be defined as a cladding breach releasing observable quantities of fission products into the coolant. Most plants are designed to operate with pin-hole size defects in up to 1 % of the fuel elements. Fission product releases above this level, either due to an increase in number or size of the cladding defects, raise coolant activity levels above those considered acceptable. In addition to the pin-hole clad ding or weld defects that could be present in the as-fabricated fuel, fuel cladding failures are generally ascribed to one of the following causes: 0 ) pellet-cladding interaction, cladding corrosion, brittle cladding failures, (4) overpressurization, cladding collapse into pellet stack gaps, cladding wear, (7) fuel rod bowing and hydriding. Fuel rod bowing will occur if the axial cladding growth due to irradiation-enhanced creep is sufficiently high so that the ends of the rod hit both the bottom and top nozzle. The rod then bows to accommodate further rod growth. This problem can be avoided by providing adequate clearance between the nozzles and fuel rods. A related problem is that of fuel assembly bow. In fuel assemblies supported by Zircaloy RCC guide tubes, differential growth of the guide tubes can cause the entire assembly to bow. Such differential growth is possible if the assembly is in an area with a high flux gradient (e.g., core periphery) and if the guide tubes show high irradiation growth rates. The growth of Zircaloy that has been an nealed sufficiently to recrystallize is generally small enough so that this does not occur. However, this problem has been observed with stress-relief annealed tubing used for guide tubes. 174 Cladding wear failure can arise if the grid-springs holding the rod in place become loose enough for the rod to vibrate. Irradiation-enhanced creep of the springs holding the rod leads to a significant stress relaxation and hence a significant reduction in spring loads when Zircaloy springs are used. The initial spring load must be sufficient so that the reduced load is still adequate to retain the rod in position. Some assemblies designed for very high burnup use Inconel springs to avoid the problem. Wear or vibration failures have also been seen in fuel rods adjacent to the core baffle in plants in which there is coolant downflow between the baffle and core barrel. Since the baffle plates are bolted together, water gets through the cracks and strikes the outer fuel pins. The jetting from this leakage can cause vibration of adjacent fuel rods. The wear failures caused by this vibration can be eliminated by locating extra rod support clips in the axial region where such jets may be encountered. 1 75 Alternatively, up flow can be used in the baffle region. Since this leads to a much lower pressure differential, jetting is largely eliminated. During the 1 970s it was found that some low-density fuel pellets den sified with exposure to such a degree that axial gaps in the fuel column occurred. Upon cladding creep-down, the cladding ovality in the axial gap
(6)
((25))
(8)
(3)
1 99
F U E L E L EM E NTS
region became sufficient for the cladding to collapse. The cladding ductility is not sufficient to allow the collapse without cracking, thus producing a failure. Failures of this type are no longer a problem since all fuel manu facturers now produce somewhat more dense fuel with a stable porosity. The use of denser, stable fuel, to prevent gap formation, and pressurizing of the fuel rods, to decrease the cladding creep-down rate, have effectively eliminated this failure mode. Overpressurization by fission gas release will not by itself lead to clad ding 'breach under normal operating conditions. However, if the internal gas pressure were sufficiently high to force the cladding away from the fuel, fuel temperatures would substantially increase. This in turn would lead to high fission gas release fractions. Such releases lead to an increased concentration of volatile fission products, which can chemically attack the internal surface of the cladding. The increased presence of these products increases the likelihood of stress-corrosion cracking. After fuel-densification failures were observed, fuel rods were pressur ized during fabrication with helium at about psi. This resulted in an internal pressure of about psi at operating conditions. While this pres sure decreased creep-down rates, it substantially increased the internal pressure at the end of life (EOL). Now that fuel densification has been greatly reduced, creep-down is seen as less of a problem. Fill pressures are often set at psi or less. The reduction in EOL gas pressure thus pro duced allows higher fuel burnups to be achieved. Although unirradiated Zircaloy can undergo significant elongation (strain) prior to failure, the irradiation embrittlement of Zircaloy, coupled with hydrogen pickup during corrosion, substantially reduces its ductility. A brittle failure will result if the failure strain is exceeded . Because of this, it has usually been assumed that the maximum strain-range seen at EOL must be limited to a small value (e.g., to prevent brittle failure. How ever, more recent experience with high burnup fuel indicates that this low a strain range limit may be unnecessarily conservative. If the ductility of the cladding is sufficiently reduced during reactor ex posure, the cladding will become brittle during cold shutdown. Fuel fail ures may then occur during the fuel handling required in refueling operations. Cladding corrosion only becomes a problem for high-burnup fuels. Cor rosion thins the cladding and therefore increases the applied stresses. In addition, the increased thickness of oxide on the surface of the cladding increases its temperature, further accelerating the corrosion rate. The hy drogen absorption of the Zircaloy is, at least to some degree, proportional to the degree of oxidation since some of the absorbed hydrogen comes from the oxidation process. This additional hydrogen pick-up further reduces cladding ductility. Proprietary zirconium alloys, which show corrosion
350
900
250
1 %)
200
T H E RMAL. A NAL. Y S I S O F PWRS
rates significantly below that of Zircaloy-4, particularly at long exposure times, have been developed in both the United States and Western Europe for use with high-burnup fuels. An accelerated corrosion rate leading to cladding failures at modest burnup can occur in the presence of heavy buildup of deposits on the ex terior cladding surface. This can occur with poor control of water chemistry but this problem appears primarily with BWR rather than PWR fuel. It has been established that, for PWR fuel, the external oxidation of the cladding should not exceed about 1 00 Il-m to avoid clad failure. 1 76 If the fuel pellet contains a hydrogenous contaminant, usually moisture, decomposition of the contaminant provides hydrogen, which can react with the internal surfaces of the cladding. It has been found that this reaction is often greatly enhanced at particular locations leading to massive local hy driding. Cladding failure generally follows. Hydriding failures were once a fairly common cause of fuel rod failure but now are relatively rare. Current manufacturing procedures appear to effectively remove any hy drogenous contaminants. Pellet-cladding interaction, or PCI, failures are the most common cause of fuel rod failure and are of the greatest concern to fuel rod designers. PCI failures can be attributed to either 0) mechanical overload of cladding sub ject to tensile stress concentration or stress-corrosion cracking. The last mechanism is recognized as the predominant failure mechanism. 1 77 Stress-corrosion cracking occurs when high stress levels are present and halogens are available at the inner surface. Cracking will occur only after the stress exceeds a threshold value and the halogen concentration is above a critical level. The halogen concentration is provided primarily by the io dine in the fuel fission products. The high stress levels are provided by pellet-cladding contact pressure. The threshold stress level for the onset of cracking decreases with irradiation exposure. 1 75 The degree of stress corrosion cracking increases with increasing stress levels. High stresses can be produced in fuel element cladding if the fuel has been operating at less than full power and then been brought rapidly to a higher power (power ramp). If the pellet-cladding gap was closed at the low power, the only way the cladding can respond to the rapid thermal expansion of the pellet is via elastic expansion. Such an expansion requires a high tensile stress and contact pressure. The maximum cladding stresses produced are considerably higher than would be calculated on the basis that the cladding tube remains cylindrical. A major reason for this is the fact that during irradiation the pellets tend to form an "hourglass" or "wheatsheaf" shape. That is, the diameter of the pellet at the axial centerline is smaller than at the pellet ends. As the cladding creeps down on the pellet, ridges are formed at the pellet junctions. At each ridge there is a significant stress concentration (Fig. 2.24). Additional regions of stress concentration occur where the cladding must bridge a crack in the fuel pellet.
(995)
(2)
201
F U E L E L EM E N TS
AXIAL FORCE DISTRIBUTION
Fig. 2.24
POlolER DISTRIBUTION
Axial force distribution on cladding wall [from C. Kjaerheim and E. Rolstad, Report, Halden Reactor Project, HPR-I07, Halden, Norway (1969)].
OEeD
When power ramping results in stress-corrosion cracking, there often appears to be a thermal feedback effect involved. The proposed sequence of events iS 1 78: 0 ) fuel with considerable inventory of stored fission gas is ramped, increased fuel temperature produces increased fission gas re lease, fission gas decreases conductivity of the fill gas (kFG=O.05kHe) and decreases gap conductance, and (4) fission gas release is further increased. This sequence can lead to substantial increases in fuel temperature and hence greater thermal expansion and higher thermal stresses. This condition is particularly likely where fuel fragment misalignment impairs the im provement of gap conductance with contact pressure. Rods that are not prepressurized are most sensitive to thermal feedback effects. By prepres surizing the rod with several atmospheres of helium at room temperature, the thermal conductivity of the fuel gas is much less sensitive to the amount of fission gas released and thermal feedback effects are greatly reduced. Improved plant operational strategies are generally successful in pre venting PCI failures. In the so called "RSST" approach179 all of the four "predictors" must exceed a critical value for a failure to occur. Minimum or critical values have to be exceeded for 0 ) the power Bange (operate at a power above a burnup-dependent failure threshold) or (2) the power �tep (siz e of power increase) or the �peed of the power increase or (4) the rime at high power. If any one of the "predictors" is subcritical, failures should not occur. The development of these criteria were based on a very extensive test program carried out at the HFR reactor in Felten, Holland.
(3) (2)
(3)
T H E RMAL A N A L Y S I S Or PWRS
Yaung ana lyzed over 7 0 0 irra d i a tions and d eveloped cu rves tha t simu lta neously fi t 80C;; of the fa i l u re and nonfai lure d a ta . conclud ed tha t the best predic tor o f the fa i l u re bou n d a ry was the the power increa se. The a l lowable power cha nge was fou nd to be a fu nc L l ! ) of burnup and ini tia l power level . Figu re 2 . 25 shows the rC I fa i l u re bou l 1 d aries they developed . 2 .5 . 3
I m p roved Fuel Rod Desi g n
I n a d d ition to establishing revised plant opera ting procedures, fu el p formance c a n be improved by i:1 nu mber of d esign mod i fications. Fu el sembly mod i fi cations lea d i ng to i mproved performa nce and h i g lk' f" bu rnu ps have been discussed previously in Sec . 2 . 1 . 2 . 3 . It has also 11l'l' previously observed that prepressu riza tion of fu el rod s with heliu m duces thermal feedback effects a n d d ecreases the probability of PCI fcl i l u fl In a d d i tion, the prepressu riza tion reduces cla d ding creep-down, thus d ucing the total stra i n range to which the cl a d d i ng is subjected . Fo r given EOL burnup, there is an opti m u m level for initial pressuriz a tion . improved fu el beha vior must be balanced against the l im i ta tion im po<"l'L by the EOL pressure of the hel i u m itself. PCI failures are also red uced by use of barrier coa tings on the inner \ \ ,; 1 1 of the fuel element cla d d i ng. The first of these barriers was a colloiLt t l d ispersion of graphite as a cladding coa tin g i n CANDU reactors . 1 K 1 Fuel fa i l u res d u e to power ram ps were sign i fica n t p roblems in the C AN D U - ty pe reactor where a t-power refueling can impose rapid power changes on t l i fue l elements being moved . The thin, col lapsed clad d i n g u sed in CAN D L reactors leads t o a particularly high stress . A t a given burnup level, therL' is a range of stresses above which fa ilu res a re likely The s tress range which fa ilures begin d ecreases as burnup i ncreases. The use of a gra p h i te barrier coa ting substan ti ally i ncreases the threshold stress. I t was origini:1 l h thought tha t the graphite acted b y d ecreasing friction between the fu el and clad d ing a nd thus decreasing mechan ical stresses. I t is now believed th ,l t the improved behavior w i th the graphite coa ting is due to the preven tion of halogen fission prod ucts from reaching the Zircaloy and thus preventing stress-corrosion cracking. PCI fai lures have also been a major problem i n BWRs where control rod motion can subject fuel adjacent to the rod to a severe power ram p . The problem has been grea tly ameliorated by the use of Zircaloy cladding lin ed with pure zirconium or zirconi u m alloyed with a small amount of i ron . p.; :' The i nner layer is coextruded with the Zircaloy and metallu rgical l y bond eLi to i t . The total wall thickness (clad ding plus l iner) equals that of con ven tional cla d d in g . The l iner provides a soft d u c tile i n terface between the p ell d and the cl a d d i ng. Because of its high d uctility, the liner itself h a s a v e n low susceptabi l i ty to s tress-corrosion cracking. The l i ner a lso serves (
-' [ M [ N TS - -- --
14
12
A B C D E F
10
� ...... '+-I
"
�
�
I
(KW 1 f t ) 4 6 8 10 12 14
8
A
6
B C
'-"
�
4
E
2
o
Fig.
2 . 25
I
Pj
------"- _�----_ LI -�-------,---, �.----,I_---I -,----
o
5
1 0
15
__
20
�� I _� i ___ _
25
B u r n u p ( GWd l to n n e U )
30
Pred icted change-in-power U�. r) failure bounda ries as a function of burnu p [from Ref. 1801 .
T H E RMAL A N A L Y S I S OF PWRS
204
barrier between the halogen fission products and Zircaloy base. While most PWR fuel manufacturers have not felt it necessary to go to zirconium-lined cladding, some manufacturers are proposing its use for fuel elements sub ject to extended burnup. Zirconium barriers, which range from to 30% of the wall thickness, have been proposed. Analyses48 indicate that the very ductile liner will carry no load and that the higher stresses in the Zircaloy lead to creep rates to above those of conventional cladding. Since PCI failures are exacerbated by high halogen concentrations, im proved fuel performance can be obtained by operation at lower peak pow ers. Some high-burnup designs call for operation at reduced linear hea t rates. In addition to ill1proved PCI performance, the lower fission gas re leases mean low gas pressure at the end of fuel life. Because pellet "hourglassing" is a major cause of stress concentration in PWR cladding, reduced hourglassing will further reduce PCI failure rates. It has been found that this can be accomplished by reducing the pellet length-to-diameter ratio. The use of annular pellets is another way of reducing cladding stresses. The free volume at the pellet center allows some of the pellet swelling and thermal expansion to be taken internally and thus reduces cladding strain and stress. Further, the reduced temperature provided by such pellets re duces fission gas release and the central free volume reduces pressure buildup. Although extensive tests of fuel rods with annular pellets have shown their value,I R3 no Western core has yet used this design. How ever, annular pellets have been used in the Russian VVER reactors. Most designs for high-burnup fuel would use cladding with improved corrosion resistance (see Sec. Such designs also tend to use the improved cladding material for the support grids and control-rod guide tubes.
10
15 70%
(1993)
2. 1 .2. 1 ).
2 .5 .4 2.5.4.1
Evaluation of Fuel Performance Du ring Normal Operation Computer Programs for Fuel Element Design
The complexity of the calculations required to treat fuel and cladding deformation and restructuring, as well as fission gas release, obviously requires that these calculations be performed via computer. As noted in Sec. the fuel is divided into a series of axial segments and radial rings within each segment. The behavior of each of these fuel volumes is then modeled over a series of timesteps so that the behavior over life may be determined. The fuel power at each timestep is set so as to simulate the expected time variation in power. The relative power in the various axial segments is set on the basis of the expected axial power variation. To obtain peak stress levels it is important that the power ramps to which the fuel may be subjected be simulated at appropriate points during the exposure.
2.4 . 1 ,
F U E L E L E M E NTS
205
Timesteps need to be adjusted to allow such ramps to be adequately simulated. A fuel-life simulation with an appropriate fuel element design code will d etermine at least the following: ( 1 ) maximum strain and strain range, (2) internal gas pressure at the end of life, and (3) maximum cladding stress. These computations are generally made assuming the cladding retains its cylindrical shape. The U.s. NRC Standard Review Plan 1 84 indicates that the uniform strain of the cladding should not exceed 1 %. Some investigators replace the simple maximum strain range limit by a more sophisticated rule. Verbeek and Hoppe47 note that, at a given constant temperature and stress, there is a maximum strain at which rupture will occur. They there fore assume that the strain produced at each stress level to which the rod is subjected consumes a portion of the total life of the cladding. If it is assumed that the applied stress variations over time is approximated by a series of time steps at a constant stress, a total damage fraction, 0' is de fined as I � Epi l 0' = L (2.202) (EL )i where I � Ep i I is the absolute value of plastic strain (less irradiation creep, which is taken as nondamaging) accumulated during the i'th time period; (EL ) i = (E )i (R), the ductile strain limit that applies during the i'th time pe riod; (Eu)i is the uniform ductility limit for the material condition (temper ature, degree of corrosion, fast neutron fluence) prevailing in i'th time period; and Ri is the factor dependent on strain rate and temperature. Fail ure is assumed to occur when 0' equals unity. Other investigators set the design limit as a fraction of the rupture strain. The circumferential strain at cladding rupture, E R , depends on the cladding temperature, degree of cold work, and neutron fluence. For the case of no circumferential temperature variation, Hagrman et al. 34 have correlated ER by where T
FR Few wt
=
cladding temperature, K
exp( - t / 1 0 23 )] / 3 = cold work function = [exp( - 21 C,) + 0.33] / 1 .33 C = fraction by which cross-sectional area is reduced by cold work neutron fluence, n/m2 = irradiation factor = [1
+2
The effect of cold work and neutron fluence on rupture strain are substan tial. A sample with 20% cold work and a neutron fluence of approximately
T H E RMAL A N A L Y S I S O F PW RS
206
1025
(2.10%203)
n / m2 would show a rupture elongation of only about of the value for unirradiated and unworked Zircaloy. Note that Eq. does not include the effects of hydrogen absorption, which further reduces duc tility, and stress annealing, which reduces the effect of cold work and increases ductility. To ensure an adequate margin of safety, the maximum design strain should be set substantially below the rupture strain (or A t the temperatures encountered i n normal operation, the cladding will be fairly brittle at the end of life and cladding failure may be expected at the rupture strain (£L = £, ). However, at the higher temperature of some tran sients, ductile behavior is encountered. With ductile behavior, failure would be encountered when the cladding strain is such that instability is encoun tered. Hagrman indicate that for K, this occurs when the strain reaches one fourth the rupture strain computed with FR = Few = As previously observed, the usual internal gas pressure limit is that there be no cladding lift-off at full power at the end of life. Maximum cladding stresses (ignoring hourglassing) are set at levels such that PCI failures are rare. Roberts and Gelhaus1 85 indicate that the threshold stress for PCI fail ure is approximately MPa. A typical design stress is about of this value. 1 86 One of the earliest computer programs to consider fuel element design in a sophisticated manner was the CYGRO code.1 66, 1 87 Both fuel and clad ding are divided into a series of concentric rings linked by continuity laws of force and displacement. The stresses, strains, and all properties are taken to be uniform throughout a given ring. During each time increment, strain rates and stress are taken as constraints in a given ring. Strain rates and stress are related by Prandtl-Reuss equations that are modified to include the effects of fuel growth, thermal expansion, plasticity, and creep. The size of a time step is limited so changes in stress and strain are small. Fuel swelling is computed via Nichols and Warner's BUBL model. 95 Po rosity migration is treated by Nichols' method. 28 The total solid volume in each ring is held constant so that the radii of the boundaries follow the fuel. The heat generation rate in each ring is a constant fraction of the total heat generation throughout time history. A comprehensive stress-strain analysis of the fuel is conducted. In the low-temperature region, allowance is made for the fact that if tensile stress exceeds a predetermined amount, the fuel cracks and tensile stress drops to near zero. Cracks are assumed to heal during compression. Frictional interaction between fuels and cladding is also considered. The finite ele ment technique is used to solve stress-strain relations. Although CYGRO conducts a very comprehensive analysis of the stress strain relationship in the fuel, it considers only a single axial node. Fission gas pressure information as a function of time must come from another calculation.
(2)
(1)
D' < 1).
et al.34
190
T < 1090
1.0.
70%
F U E L E L EM E N TS
207
The LIFE code1 H8 eliminated the need for obtaining fission gas release separately, since it is capable of considering a series of axial nodes at dif ferent power levels. This code was originally developed for fast reactor fuel, but versions satisfactory for use with LWR (SST 1H9 and LIFE-THERMAL) are available. The stress-strain analysis performed by the LIFE code is less elaborate than that done by CYGRO, since only three fuel regions (colum nar, equiaxial, and unrestructured) are considered in the stress-strain anal ysis. The swelling and fission gas release analysis is more sophisticated since the GRASS code96 is used for computation of these quantities and a number of axial nodes are considered. Both CYGRO and LIFE are time consuming to run. In the first generation of easily usable design codes, approximate stress analyses are conducted and swelling and fission gas release are described by analytical expressions. The COMETHE (Ref. 91), FRAP-S (Ref. 1 90), and GAPCON (Ref. 1 9 1 ) codes are examples of this approach. In all of these codes, a series of axial seg ments is considered so that a reasonable estimate of overall fission gas release can be obtained and its effect on cladding and fuel behavior can be determined. A second generation of design codes, which use improved correlations for fuel and cladding behavior but which also perform only a relatively simple stress analyses, are now (1992) available. These include COMETHE (Ref. 192), ESCORE (Ref. 48), TAC03 (Ref. 1 93) and other reactor vendor codes. A number of the more recent fuel element design codes continue to use relatively simple approaches for fission gas release and fuel swelling but perform fairly rigorous stress-strain analyses. Some of the better known programs in this category are: BEAF (Ref. 1 69), SAMURA (Ref. 194), ENIGMA (Ref. 1 95), FEMAXI-III (Ref. 1 96), FRAPCON (Ref. 120), SLEUTH (Ref. 1 97), and TRANSURANUS (Ref. 1 98). The FREY code (Refs. 1 99 and 200), which uses a fully coupled two-dimensional finite element analysis, differs from the other programs in this category in that it can consider azimuthal variations. In addition to the LIFE codes, several programs now incorporate fun damental models of fission gas release and fuel swelling and conduct a sophisticated stress analysis of fuel and cladding. FUTURE (Ref. 201 ) and WAFER-3 (Ref. 202) are examples of programs of this nature. 2 . 5.4.2
PCI Fuel Failure Estimation
The computer programs for fuel element design discussed in the previous sections allow the designer to determine whether his or her design exceeds any of the accepted thresholds. However, even though all design criteria are met, experience indicates that a small number of fuel failures will still Occur.
T H E RMAL A N A L Y S I S OF PWRS
2 08
Rolstad 203 argued that stochastic elements play an important role in de termining power range failures. He concluded that failures occurred when controlling parameters were at the most unfavorable ends of their tolerance bands, or when the most unfavorable fuel crack pattern coincided with the worst cladding defect. Therefore, he proposed a probabilistic approach. In Rolstad's original POSHO code,203 local power ramps resulting from planned fuel operation are simulated. These are then processed through a fuel failure probability routine. Failure probability estimates are based on fuel design parameters, burnup, and previous power ramp failure experi ence. The failure probabilities for a given power change are a function of local power level. The number of cladding cracks produced by the change is obtained by multiplying the number of pellet interfaces at a given power level by the average failure probability at that power level. In the original POSHO program, the concept of "power shock" was used to represent cladding stresses resulting from rapid power increases. The "power shock" at any time is the difference between the real nodal power and the power corresponding to the zero gap and no contact pressure. In the newer POSHO-THERMAL,204 fuel temperatures and fission gas releases are computed. The concept of "temperature shock" (actual temperature less temperature for zero contact pressure) is now used as a stress index. This results in more accurate failure predictions since fuel temperatures are not always directly proportional to power (e.g., large fission gas releases de crease gap conductance). The SPEAR program 101 also allows the examination of structural patterns of fuel failure. Through application of a pattern matching program to a large base of fuel failure data, failure probabilities are assigned to regions in which various operating conditions (e.g., burnup, maximum fuel tem perature, contact pressure, iodine concentration, ramp rate) or combinations of these conditions are within a given range. The SPEAR program then compares the time-varying operating conditions to these failure probability regions and assigns appropriate failure probabilities to the design core. More mechanistic cladding failure models are also available. In one ap proach, 1 O I the mean times to failure obtained from stress-corrosion cracking experiments are correlated as a function of maximum hoop stress, iodine concentration, and cladding texture angle. A damage parameter, is then defined as the ratio of the operating time under given conditions to the failure time. The total damage is the sum of these ratios. The fraction of failures for a given total is obtained from a log-normal probability dis tribution derived from a fit of the laboratory cracking data. In using this correlation the cladding stress is based on an estimate of the maximum stress at the cladding ridges. Yaung af. 1 HO have developed a more elaborate stress-corrosion cracking model. They divide the fuel cladding into a number of radial elements and keep track of ( 1 ) the local stress-strain level (allowing for effects of ridges
d,
d
et
F U E L E L EM E N TS
209
and cracks) in each element and (2) the iodine environment in the first (innermost) unfailed radial element. Failure of a given radial element is determined by an experimentally established criterion. 2.6
TRAN S I E NT F U EL ROD B E HAVI O R
2.6.1
Fuel Rod Temperature Transients
2 .6 . 1 . 1
Ana lytical Techniques
Under transient conditions when heat storage must be considered, the gen eral conduction equation takes the form
aT V(kVT)
cpP ill =
t
+ q '"
(2.204)
where represents time, cp specific heat, P density, and other symbols have their previous meanings. We consider a fuel pellet of radius surrounded by cladding with an outside radius If we ignore axial conduction and assume that for the fuel is a constant, for the pellet region we write
a b: k1 ( aar2T2l 1r aTlar ) '" - cP1 Pl aTlat (O
+-
-
+q
-
k
(2.205a)
while for the cladding region +
=C
(2.205b)
t
We consider a step change in heat transfer coefficient at the outer clad ding at = 0 and a subsequent reduction in heat generation rate. This cor responds to the situation arising during a transient when the fuel element is suddenly blanketed by steam [departure from nucleate boiling (ONB) occurs (Sec. 4.4.2)] and the film coefficient is drastically reduced. Heat gen eration rate is reduced on reactor scram. A set of dimensionless groups is defined as follows: (2.206) where hg is gap conductance and
h
( k2 ) at2
is the film coefficient at rod surface T=
(2.207)
Cp2P2
K(R kl(r) 1 ) = k2
'
(2.208)
T H E RMAL A N A L Y S I S O F PWRS
210
+ k2T T = q "'(0)a 2
(2.209)
-
(
)
Then Eqs. (2.205a) and (2.205b) can be written as K
aTt a2 Tt + 1 aTt + - -- CP -Q aT aRi Rl aR 1 aT2+ a2 T2+ � aT2+ = + aT aRi R l aR1
--
( )
(2.21 0a) (2.210b)
The boundary conditions are
aTt aR1 + + _ BT/ B 1 ( T1 - T2 ) - -BRI
+ B 1 ( T1+ - T2 ) =
-
K
(2.21 1 )
-
+ + aT/ B 2( T2 - T ) = - aR 1 %
(2.212) (at R1 = 1 )
(2.21 3)
where T:; is the dimensionless bulk coolant temperature. These equations can be solved by a Hankel transform20s or a finite in tegral transform. 206 In both cases, the exact solutions are in terms of infinite series. By using a finite integral transform technique, Matsch206 obtained the following solution:
T + (R 1 ,T) = I CI1(T) Xn ( R 1 ) /1
(2.214)
where values X/1 are obtained by considering an associated equation of the form (2.21 5) to which the solutions are a set of eigenfunctions ( Xn) true for discrete values of (�/1 )2 A complex solution procedure is required to determine the eigen functions. Because of the complexity of this procedure, and because analytic techniques do not provide good accuracy when large temperature changes make the assumption of constant properties invalid, analytic techniques that attempt a complete solution of the problem are now rarely used. 2.6 . 1 .2
Simplified Ana lytical Techniques for Temperature Transients
Analytical techniques, which are less tedious than described in Sec. 2.6.1 . 1 , can b e useful i n some instances. They can provide a rapid means for ob taining approximate estimates under many conditions. In addition, for rel-
F U E L E L EM E N TS
atively slow transients, the results obtained can be adequate and not require a more sophisticated procedure. The loss-of-flow accident (LOFA) with scram sometimes falls into this category. For a V02 fuel rod, maximum cladding temperature occurs several seconds after initiation of the accident and a lumped parameter procedure yields a reasonable estimate. Lumped parameter techniques are among the earliest procedures used for examining temperature transients. 207 In the lumped parameter proce dure suggested by Tong,20B thermal resistances and the capacitances of the pellet and cladding are evaluated at their average condition in time and space. Each quantity is lumped at the middle of the physical geometry and axial conduction is neglected. Again, we consider the situation where there is a step reduction in the coolant heat transfer coefficient and a subsequent reduction in heat generation rate due to reactor scram. Rate equations for heat transfer from the U02 fuel rod can be written as
dT1 T1 -T2 R
q ll, = C l Tt + ---
(2.216)
1
and (2.21 7) where
q � = nuclear heating, Btu / s ft of fuel rod C = thermal capacitance of pellet, BturF ft, C = TIrlCp P l 1 1 1 C2 = thermal capacitance of clad, Btu rF ft, C2 = 2TIr2 (�r)cp2 p 2 for thin cladding R l = resistance of V02 and gap, s ft of/Btu = [1 / (8TIk1 )] + [ 1 / (2TIr 1 hg )], where kl is U02 thermal conductivity and hg is gap conductance R 2 = resistance of coolant film = 1 / (2TIriz), s ft of/Btu Tl = average pellet temperature, of T2 = average cladding temperature, of = bulk coolant temperature, oF.
Tc
Tc T/t).
t
After a pipe rupture, the system pressure and saturation temperature of the coolant drop with time. Hence, = Time is counted from the instant of rupture. From the Laplace transformation of Eqs. (2.216) and (2.21 7), we get
T H E RMAL A N A L Y S I S OF PWR S
(
1 )
1 )
(
R2 R2 R2C2S + + � Rl q n (s) + Tc (s) + R2C2S + + � R I CI TI ( O) + R2C2 TiO) I
2 R lR2Cl C2S + ( RI CI + R2C2 + R2CI )S +
1
(2.219)
By knowing coolant temperature and fuel rod power as a function of time, the histories of pellet and cladding temperatures can be compu ted from the inverse transformation of Egs. and In the analysis of loss-of-flow transient, two simplifications can be made: First, when the pumps lose power, system pressure does not change sig nificantly and coolant temperature Tc remains approximately constant; sec ond, maximum cladding temperature usually occurs within s of the instant of loss of power to the pump. Hence, decay heat can be assumed constant during this short time period . Our boundary conditions now become
(2.218) (2.2 19).
10
for t :::; 0:
q ;l = q ;/ O)
R2 = R2,o
for 0 < t :::; t\ : q ;l = q ;/O )
R2 = R2,
for t � t 1 :
R2 = R2,
q:l = j3 q ;l ( O )
(2.220)
fi l m boiling film boiling
(2.217)
By taking advantage of the constancy of Tu we can solve Eg. for T] and then differentiate with respect to t to obtain dTI / dt. We then rewrite Eg. as
(2. 2 16)
(2.221 )
where 8 = T2 - Te ' When the Laplace transform is taken, we obtain
� [1 -
� exp (t , s) ]
=
[
C, CzR, Osz - 9(0)s -
8
d
��o) ] (
+ C , + Cz +
X [ 8 s - 8( O) ] + R2
where e represents the Laplace transform of 8. The value 8(0 ) is given by 8(0) = q:l(0 )R2 o ,
and d8(0 ) /dt from the relationship Tl (0) = q �(O )(R l + R2,o)
C��z )
(2.222) (2.223) (2.224)
F U E L E L E M E NTS
21 3
and Eq. (2.2 1 7) . Substitution of these numerical values, followed by the inverse transformation, yields e as a function of time. Chen extended the lumped parameter technique to consider phase changes (cladding and fuel melting). They considered the problem of loss of flow at constant power (failure to scram) and developed equations for estimating the times at which cladding and fuel melting begin. Steady state conditions are computed first, assuming the coolant is at saturation. The average cladding temperature, Teo ' at steady state is
et az.2°9
1 1 1 ri q lll rf --� + Tc,, = Tsa t + -2 h a rc kc 2 rc rF
(
[
-
-
and the average steady-state fuel temperature,
rc
In -
TFo '
)]
rF
is given by
1 1 1 r q lll r 2 1 + + - ln� TF = Ts t + ---...I - + 2 rF 4kF h a re h ga pr F kc " a
(
-
(2 .22 5)
)
--
(2.226)
where ho = q lll = rF = rc = kC 'kF = Tsa t =
steady-state (pre-DNB) coolant heat transfer coefficient volumetric heat generation rate outer radius of fuel region outer radius of cladding thermal conductivity of cladding and fuel, respectively saturation temperature.
They then assume that the coolant heat transfer coefficient goes to zero (va por blanketing) and remains constant at that value. After defining the fol lowing symbols, they develop the working equations shown in Table 2.VIII, where
td
=
Pc,cpC /Hc = PF,cPF,HF = h'
time at which vapor blanketing (DNB) occurs density, specific heat, and latent heat of cladding, respectively density, specific heat, and latent heat of fuel, respectively
(
CF = rFPpCPF/ 2h ',
tern pera ture
C� Ce
Q
)
rF 1 rC - rF + + 4k � F h ga p (
1
, aver��e overall heat transfer coeffICIent calculated at the steady-state maximum fuel
= rFPFcpF/ 2h ' , calculated Ta v = [TFP + TF ( tm ) 1 / 2
[ (rL� - rf ) Pccpc1 /2RF h ' , Tcl' lll = rFq / 211 ' .
temperature,
at average temperature
calculated at the cladding melting
Clad melting phase
Transient phase before clad melting
Phases
-
t,, +
1/1
-
l' +
t + l'
t
TFO - T, 0
TFO - Tc o
---
,
(r} - rl )p,H,
_
Cc
for
2rF h ' (TFU - Tc )
(r} - rl )p,Hc
Tcl ' - Teo
(
Cc
CF + Ce
)
TFU - Tc o
Cl + -CF + Ce
c)
, 2RF h (TFO - T
(r} - rl ) P eHe
tl' - td > - CF + C c
c F cc
2 rF h (TFO - Tc )
( r} - r l ) PPc ,
C F Cc
t ' - td > l
-
t" <S CF
III - tl ' > C F
111
CF + Cc
-
CF C,
( CF + C,. )
t" - td <S -C F + Cc
C V + Ce
for
+ c F ---
CF Cc t" - td <S --CF + Cc
2R F h (TFO - Tel')
Case 2: for
111
t,, +
Case 1 : for
1 1
"
< td +
td +
Tcl ' - T,- o
(complete time of clad mel ting)
t,, =
,.
td < t <S tp (starting time of clad melting), where
( ) [Tcl' -T,o (�)2 ]
Timing
Q
Cc
- td <S
CF Cc
C F + Cc
{
-
C r Ce
C
(T
�()
)
{
F=
-
CF Ce
CF + Ce
C1
for
for
TFO + Tel ' -
TFll - T, o ( Cc + t - t ' ) Tel + , , CF + Cc
l'
t - td >
Tro + Tel, -
Case 2: for
TF -
t - tl '
C F Ce t" - td <S -CF + Ce
TFO + ( Tel' - Tc ,,) --
Case 1 : for
Q
- Te ) + -- ( t - t d ) ,' C F + C, .
C F + C,
� -+ t - td
( C F + c,)2
(
t - t l ' <S CF t - tl' > CF
for for
t - t,, > CF
t - t l ' <S CF
Tc = Tcl' (clad mel ting temperature)
FO
Q
CF Cc t - t - CF + Cc
+ -C v + Cc
Case 2: for
T, = T" , + - ( t - td) and TF =
Case 1 :
Average Fuel and Cladding Temperatures
TA B L E 2.V I I I Working Equations for the Chen e t al.209 Transient Cond uction Model with Phase Change
�
;;;::J V1
""tI
'T1
-< V1 Vi o
» z » r
r
»
3::
m ;;;::J
..., I
Fuel melting phase
Transient phase before fuel melting
"
�
t 11 /
+
tll/ +
I
tpF +
F'
for
� �;
rF PFHF
211 , (TFO - Tc )
+
TFo - Te
Cc[TFP - Tc(tPF )]
]
for
for
� t PF + Cc
t Fill > t PF + Cc
t Fill
C� + Cc
C�- C(
CF + C(
CC
tpF - t lll >
(TFO - Tc ) / ( C� + CJ
F PF 2h-'---+_--' " -'__-'-p + Tc(t F) ] [ TFO T ....:. P
� ��
t
[
I"
T, I ' +
�
1
TF (t '/I ) -
TF (t/l,) -
"
{
[TF (t/l,) - T'I,](t -
TF = TFI'
for
t - t ill �
(
Cc
t - tPF
(fuel meltin g temperature)
exp - --
{
)
�
,.:::: for
} C� Cc t - '" > -C� + Cc
t + t 1/ / C� + Cc
for
C� Cc
C� + Cc
Cc C�t - IIl ""-" C� + Ce
t - t ill >
[ TF (t/l,) - Tel' - TFO + TcoHt - till )
TFO - Tc [, Cc [TF (t Ill) - Tcp] TFO - Tc o C� + Cc
1
C�
for
}
C� Cc C� + Cc
_ C� Cc _ TFO - Tc o C � [ TF (t ill ) - Tc) + t t ill , CF + Cc TFO - Tc o CF, + Ce
c:-
Tc = TFP - [ TFP - Tc(tP F ) ]
TF =
,.
"
Tc - L, Tel' + _
2.VI I I (continued)
C� Ce t pF - t ll/ � ' Cc CF +
Cc TF (tlll) - Tcl' CF + Cc
for
(complete time of fuel melting)
TFP - TF (tIl1) +
TFO - Tc ,, - TF (t,,) + T,p
[TFP - TF (t,n> ]C�
(starting time of fuel melting), where
t PF < t � t Fi ll
t PF =
I"
11/ < t � t pF
TAB L E
U1
N
m Z -i IJ'1
�
C m rm r m
"'T1
T H E RMAL A N A L Y S I S O F PWR S
216
2 .6.1 .3
Electrica l A na log Techn iques
The lumped parameter technique is basically equivalent to using an elec trical analog. Conditions at the nodal points of the electrical network rep resent the average conditions of a particular region. To use such an analog, we note the correspondence between voltage and temperature, current and heat flow, and the product of electrical resistance and the capacity of the electrical system and thermal diffusivity. Thus, Fourier's law of q = � T/R, is represented by Ohm's law for current flow = �E/R. The change in system thermal energy is given by
i
(2.227) where Q
heat content, Btu specific heat = volume p = density.
cp
V
The change in electrical energy is represented by dQI' dE =C dt dt
(2.228)
where Qc is the electric charge on the condenser and C is condenser capacitance. The electrical analog technique was widely used for control and transient analyses and in nuclear power plant simulators. However, the rapid ad vance in digital computer capabilities has led to the virtual disappearance of analog computation. 2.6. 1 .4
Approximate Numerical Techniques
An analytical solution of the lumped parameter equations can only be ob tained for relatively simple boundary conditions. During many accident situations, rod power and coolant heat transfer conditions vary in a com plex manner. This necessitates a stepwise numerical integration of the gov erning equations lEqs. and written in finite difference form. Although we may continue to represent the cladding and fuel in terms of single average temperatures, it is also feasible to allow for a parabolic tem perature profile in the fuel. By assuming a parabolic temperature profile, improved accuracy can be obtained with little increase in computational effort. With the parabolic pro file assumption, the pellet temperature profile may be written in terms of the fuel center temperature, TFe , and the pellet surface temperature, TFS ' We then have
(2.216) (2.2 17)]
F U E L E L EM E NTS
(2.229) is the fuel temperature at radius r and other symbols have their where meaning. In some approaches,2l 0 the cladding temperature is also previous assumed to have a parabolic profile. However, since the temperature drop across the cladding is small, it is more usual to assume a single cladding temperature, We obtain the changes in T1 and over time period by writing the fuel and cladding heat balances in finite difference form after adding a term accounting for any steam-cladding reaction. We then have
TF
T2.
T2
q
(6.t)
, + ( TFS - T2 ) = ( c2 6.T26.t ) + ( T2 -Tc)
c
R
R(
�
(2.230) (2.231)
where
RF =
1 I 87rkl =
(2.232)
resistance of fuel
and q ;/ q�
= rate of heat production in fuel (energyIL6.t) heat addition to cladding due to any zirconium-steam reaction (energy I L6.t)
outside radius of fuel pellet r = inside radius of cladding = average thermal conductivity of fuel and cladding, respectively = heat capacity of fuel and cladding, respectively (energy average temperature of fuel and cladding, respectively = change in over time period = coolant temperature. rl
=
2 k}!k2 C1,C2 T1,T2 = 6.Tj Tc
We may relate and noting
IsL6.T)
Tj
6.t
T1 to TFS and TFC by making use of the parabolic profile (2.233)
By equating the radial heat flow at the pellet boundary to the heat transfer across the gap we have
T H E RMAL A N A L Y S I S OF PW R S
21 8
- kf
dTr dr
l
r = rl
=
2 kf (TFc - TF s ) r1
(2.234)
where kf = thermal conductivity of fuel at TFS • With the use of Eqs. (2.233) and (2.234), TFC and TFS can be eliminated in Eqs. (2.230) and (2.231 ). We then are left with two finite difference equations to be solved simultane ously for T1 and T2 at time (t + L1t). A revised value of (q�), (q�), and material properties are then used to produce a new set of finite difference equations to be used for the next time period. Ghiassian et al. 2 1 0 compared parabolic profile results to computation of fuel and cladding temperatures using a series of radial nodes. They concluded that the parabolic-profile approach gave sufficiently accurate predictions for use in a LOeA. Numerical Methods for Computation of Transient Temperature Profiles
2.6.1.5
If a detailed examination of the fuel temperature profiles during a transient is desired, it can be readily accomplished by numerical techniques designed for digital computer use. In addition to accounting for the time variation in power and coolant conditions, these methods can allow for the changes in fuel and cladding properties with varying temperature profiles. Discrete time and space intervals are considered and the general conduction equa tion, Eq. (2.204), may be written in finite difference form. If axial conduction is ignored, for an annular region at a given elevation we write V}. Cp [ TJ,I. - l ] X
where
(
T},I. - T} ,I· �t . 1
1)
�+
=%
T} - l ,1' - 1 - TJ,I. . - 1 R}· -
_
TJ,I. . - 1 - T} + 1 ,t. - l R}.,}. + l
(2.235)
Vj
= volume of region j, per unit of length cp[Tj ) = volumetric heat capacity of pellet at the temperature of region j at the end of timestep i Tj,; = temperature of region j at the end of timestep i - 1 L1t; = time increment q; = heat generation per unit of time and length in region j Rj - 1 , j = resistance between region j - 1 and j computed as k [T( - l, ); - I ] (Aj - 1 ) 1 __ = p j j Rj _ rj - rj _
(2.236)
with kp [T( j - 1 , j ), i - 1 ]
=
conductivity of U02 at a temperature that is the average temperature of regions j - 1 and j at the end of timestep i - 1
F U E L E L E M E NTS
A) - 1 ,) = =
r) - 1 and r)
219
heat transfer area between regions j - 1 and j average radii of these regions.
For the last region in contact with the cladding, we write
R.j,C = X j,e 1
A
hr
-
(2.237)
.
where hr is contact conductance. The basic equation can be solved as
Tj,i Vj. q;[TIltjj., I-. 1 ] + mj - 1 ,j Tj - j - 1 + mj,j + 1 Tj 1, =
c
1,
P
+
(2.238) where
mj,j +l = V.Cp [T . 1]R. j
j, 1 -
(2.239)
J,
The foregoing formulation is called an "explicit" solution since tempera tures at new time i can be calculated directly (explicitly) given the condi tions at time i - 1 . Equations of this nature are subject to numerical instability if the timestep chosen is too large. Thus, Eq. (2.238) yields a negative (unstable) value for if
Tj, j
(2.240) Since the right side of Eq. (2.240) is always positive, we can avoid this condition by assuring ourselves that
mj -1 ,j + mj,j +1 tj t . Vlp [TjR, ji - 11,](Rj +Rj j1,,j )(Rj ,j +
(2.241 )
- 1 �O
Thus, the maximum value of a can be determined as a
Il tj
1
�
1)
(2.242)
The minimum of values determined for the several segments is used. There are two disadvantages to this procedure: First, we are computing the net heat flow into a given annular segment based on temperatures at the previous time. Second, restriction on the size of the time increment may require an excessive amount of computations for problems extending over long time periods. These difficulties can be obviated using the "implicit" form of the difference equation. Here we determine heat flows from tem perature differences at the advanced position in time. Thus, Eq. (2.235) becomes
T H E RMAL A N A L Y S I S O F PWRS
220
V.c /
P
· [T . ] (T . -T dt . /,1
X
j,1
;,1-
1
' q + ,
)
1
(T - T . ) ]
-
1 , I·
" I
R]. -
(2.243)
Initially, we assume that the specific heat and R have values equal to those at the previous time. We then have a set of linear equations, which can be solved simultaneously for If properties change significantly with tem perature, updated values for the current time are used in an iterative procedure. While the implicit method removes tight restrictions on the value of chosen, use of a large value of increases the error due to discretization. This discretization error is of If temperatures change significantly over a time stop, it is more realistic to evaluate the temperature changes using average values for the temper ature differences, properties, and rate of heat generation. This formulation, generally designated as the Crank-Nicholson approach, yields
Tj ,
L1t;
O[ddt;t; + (d r)2].
V (cp ) X . j
( T,., . ) = [ (T,. ;) + (T;. - i-I) ] dt· , T;. c '
q I. + ,
c-1
-
_
1
.
- 1
'
_
-
2R·
1 ;- 1 '
T;.
,
-
(2.244)
where
( 2 [T( ,m) ] + 2 [T(I,m) i] ) (d[,m) =------- kp
_
R[,/n
l
c
-l
kp
re - rm
q; = average value of qj
i
over the timestep.
Since conditions at time are unknown, the set of equations represented by (2.244) is solved initially using values from the previous time step for cp and R. The equations are then resolved iteratively with updated property values. The Crank-Nickolson approach is unconditionally stable and has a dis cretization error of It thus has less of an error due to the than the original implicit procedure. The foregoing finite difference procedures can also be used for calculat ing cladding temperatures by using cladding properties and setting q ' to zero at interior nodes and equal to the heat generated by the zirconi� m water reaction in the outermost region. For the node point adjacent to fuel,
dt
O[(dt)2+ (dr)2].
F U E L E L EM E NTS
R . is defined in terms of gap conductance, while for the node point adjacent t6 the coolant, Rj is defined in terms of coolant heat transfer coefficient.
Finite difference procedures, similar to the types described in this section, are commonly incorporated in the computer codes used for analyzing core thermal and hydraulic transients (see Sha 2 11 for overview of such analyses). power generation versus time is usually input and the program computes fuel, cladding, and coolant temperatures plus heat input to the coolant. Although the relative simplicity of the finite difference approach has led to its widespread use in nuclear fuel element analysis, other approaches such as the "finite element" method2 1 2 may be used. Comparisons of the finite element and finite difference methods2 1 3 at conditions typical of fuel transients have been made. The results do not show that any clear improve ment in accuracy or computing time is gained by use of the finite element technique. Lassman2 1 4 has proposed a solution technique based on the use of a quasi-steady-state approximate function:
T [r,t] = q�l l k[t] r 2 + Cdt] ln r + C2[ t] (2.245) where qrlI [t] = qlI [t] - cpP aatT = q1 1[ ' ] _ Cp ( T[r, i] - Tt [r,i- 1 ] ) The constants C [t] and C2[t] are replaced in terms of the temperatures at 1 the inner and outer boundaries of the given annular ring. For node j, this results in I I [ ( r 0 )2 ( r 1 ) 2 ] } , rOj ) l qr O Tj [r,t ] - TO k4qrI I [ (rj0 )2 -r 2] + IIn(r/ T T { X k j j 4j j - j n ( �0 / rj1 ) (2.246) 1
-
j
+
�
+
j
where the superscripts 0 and 1 refer to the outer and inner radii of node j. The nodes are linked by setting the heat flux at the outer edge of node j equal to the heat flux at the inner edge of node (j + and setting Tp = T}+ l ' The required heat fluxes are obtained by differentiating Eq. with respect to at each node. The resulting equations, representing the heat flux equalities at the node boundaries and the outer boundary, can be rearranged in the form
1)
(2.246)
r
aj
T?
- l, i
+ bj TF i +
cj To - [(T O
cjTF+
1 , i = [ ( T?- l , i - l, T?i - ]t TF+ l ,
(2.247)
when ai' bi, and are constants for a given radius and set of properties. The simultaneous equations may be solved by Gaussian elimination or other matrix techniques. Alternatively, they may be written as bj j, i -
TO TO ) TO
i - 1 , i - l ' i, i - l ' j + l, i - a j - l , i - Cj
TO
j + l, i .
(2.248)
T H E RMAL A N A L Y S I S OF PWRS
Each equation can now be solved individually by an iterative process, using values derived from the previous iteration, for the current temperature on the right-hand side of Eq. Lassman2 1 4 concludes that, in view of the necessity for updating the fuel properties as the temperature changes, an iterative solution of the set of equations represented by Eq. can be preferable to the repeated use of Gaussian elimination. The rapidity of the iteration scheme depends on the method used for assigning the tempera tures on the right hand side of Eq. Lassman finds that rapid con vergence is obtained when the (Tj , ) / II + l ' the temperature values used at iteration + iteration, be computed from:
(2.243).
(m 1)
(2.243)
(2.243).
{
} (2.249) m,
[(Tj , i ) /II - (Tj , i ) m - l ] [(Tj , i ) II1* - (Tj , i ) m ] (T; , i ) m + l = (Tj, ) m [(Tj , i ) /11* - (Tj,i ) /II ] - [(Tj,i ) 111 - - ( Tj , ) I11 - ] 0 1 1 where (T; , i )11l is the value used on the right-hand side in the iteration and (T; , i ) lII * is the value of(Tj ) obtained as a solution (left-hand side) in iteration m. The iterative solution scheme indicated here can also be applied to the equations derived from the finite difference method. All of the foregoing one-dimensional models make the tacit assumption that axial conduction is negligible. This has been found to be correct in nearly all cases. Two notable exceptions are when the cladding segment considered is adjacent to the quench front in the core recovery phase of a loss of coolant accident or is adjacent to the end of a partially inserted control rod.
(1)
2.6.2 2.6.2.1
(2)
Cladding Behavior D u ring Transients Cladding Beha vior During Rapid Power Transients
The approaches and computer programs used to analyze behavior under normal operating conditions are readily extended to rapid transients. The modifications required to accomplish this generally consist of use of a fission gas release model which incorporates transient effects (e.g., release due to grain boundary sweeping), incorporation of an axial gas mixing model, which includes axial gas flow due to pressure gradients and diffu sion, and addition of a transient heat transfer model. Some programs also include a thermal-hydraulic analysis of the subchannel in which the hot fuel rod is located. Alternatively, the fuel rod analysis program may be linked to a separate thermal-hydraulic program, which then provides the necessary coolant boundary conditions. Computer programs designed for analysis of fuel element behavior during transients include COMETHE,2 1 5 FEMAX,2 16 FRAP-T,2 17 and TRANSURANUS.1 98 .
(3)
2 .6.2.2
(2)
(1)
Zircaloy Behavior During a Hypothetical L OCA
While many early transien!t analyses were concerned with demonstrating that center fuel melting did not occur, more recent studies are primarily
F U E L E L EM E N TS
concerned with cladding behavior. It is recognized that most analytical techniques are valid only as long as a coolable geometry is maintained. During a LOCA, cladding temperature can rise to sufficiently high levels so that the hottest rods swell and burst locally with a longitudinal split. Providing the cladding is not too brittle, the cladding remains in one piece and the pellets are restrained. The design limitations established are based on this requirement. It was previously observed that Zircaloy reacts with steam at high tem peratures to form zirconium oxide and hydrogen. The rate at which this reaction occurs was originally described by the well-known Baker-Just relation2 1 8 w2 =
(33.3 06 X1
t ) exp(
-
45,000/ RT)
(2.250)
=K t p
where
W T R
= weight of zirconium oxidized, mg / cm2 =
temperature of unoxidized zirconium, K
t = time, s = universal gas constant,
1.987
K = rate constant, (mg) 2 · cm - 4 · s p
cal /mol K
Based on more recent data, Thomas and Chexl2 19 recommend the following revised values for Kp
(2.251)
( T
K = A exp B / ) p
where
A=9. 15 105103 6 AA=3. = 9.53882x10 A=33.6 x 106 X X
B= B= B= B=
800�T� 1089 10891850
-13, 74060 -21, 3 -16, -22,986000
K K K
K
The reaction rate may also be related to the distance, 0, that metal-metal oxide interface has moved from its original position by do
dt
=K X p
10 (2PZTO) 3/
2
PZT
(2.252)
where 0 is the distance the interface has moved (mm) and is the density of zirconium (kg/m3 ). The rate constants of Eq. describe the behavior when the reaction rate is limited by the rate of oxygen diffusion through the oxide layer. However, when hydrogen concentrations are high, the reaction rate can also be limited by the diffusion of steam across the gas boundary layer to the oxide surface. Under these latter conditions,
(2.251)
T H E RMAL A N A L Y S I S O F PWRS
do dt K, = Mzr = y=
yMzr Kg 2Pzr
(2.253)
where s
mass transfer coefficient, mol ! (m2 . s · mole fraction) molecular weight of zirconium mole fraction of steam in gaseous phase.
Kg,
Because of the low steam flow rates in the steam-cooling phase of a LOCA, the mass transfer coefficient, is obtained using the analogy between heat and mass transfer in laminar flow: Kg where
== P' = R= == == Pg ==
De Df
T
Nu Gr Pr Sc Tg,Tw (3
i-Lg
(2.253).
= --R g-P' e (- ) Dr
T D
Nu
(2.254)
SC V3 Pr
mass transfer surface equivalent diameter, m diffusivity of steam in steam-hydrogen mixture (m 2 / s) total pressure, atm temperature of gas mixture, K ideal gas constant, atm m3 / mol K Nusselt number (Gr Pr) l/3 Grashoff number g(3 ( Tw - Tg ) De3 p� / i-L� Prandtl number Schmidt number temperature of wall and gas mixture, respectively, K density of gas phase viscosity of gas phase coefficient of volume expansion.
0. 1 3
The reaction rate is determined by the minimum
(do dt) /
of Eqs.
(2.252)
and
The zirconium oxide film formed on the surface of the cladding acts as a thermal barrier. At the same time, zirconium metal is consumed, effec tively reducing the wall thickness of the cladding. In addition, Zircaloy tends to dissolve its own oxide, producing a cladding with increased oxy gen content and decreased ductility. Earlier we noted that at temperatures above K, alpha-phase zir conium is transferred to the beta phase. At temperatures where zirconium oxidation takes place (between and K) the outer surface is oxi dized to Zr02 • This is adjacent to a region of alpha-phase zirconium sta bilized with a high concentration of oxygen and then a region of mixed
1400 1700
�
1040
F U E L EL E M E N TS
alpha and beta phases inside this. The inner region is likely to be pure beta phase material. The oxide and alpha-phase material are brittle; ductility of the cladding is primarily due to the beta phase. The embrittlement of oxidized Zircaloy has been studied by Hobson and Rittenhouse22o and Hobson. 22 1 Hobson and Rittenhouse22o initially con clud ed that the ductility of Zircaloy depended on Fw , the fraction of original wall thickness remaining in the beta phase at operating temperature. How ever, Hobson22 1 subsequently concluded that ductility depends not only on Fw' but also on the amount and distribution of oxygen in the beta phase. These latter quantities appear to be a function of the oxidizing temperature with increased oxygen dissolved at higher oxidizing temperatures. In view of these facts, the acceptance criteria for ECCS for water-cooled reactors222 require that total oxidation of the cladding be limited to a maximum of 1 7% of the original cladding material and that the maximum cladding tem perature during the transient be limited to 2200°F. Therefore, analyses of a hypothetical LOCA must demonstrate that neither of these limits are exceeded. Metallurgical interaction between Zircaloy and stainless-steel or nickel based alloys used as spring or spacer grid materials can occur at temper atures in excess of 900°C. The eutectic temperatures for the possible alloy systems223 are Zr-Fe, 934°C; Zr-Ni, 961°C; and Zr-Cr, 1 300°C. Alloying is apparently inhibited to some degree by crud and oxide layers. Pickman223 notes that United Kingdom Atomic Energy Authority experiments show that Zircaloy-2 can braze to stainless-steel grid ferrules at temperatures in excess of 1 200°C. The 2200°F (1204°C) temperature limit set by U.s. emer gency core cooling acceptance criteria effectively serves to limit possible alloying. Cladding failure can occur when internal gas pressure causes cladding ballooning and subsequent rupture at local spots. It is of interest to be able to calculate the degree to which such failures are possible under various hypothetical conditions. Several computer code models, such as SST (Ref. 1 89) and FRAP-T (Refs. 224 and 225) have been designed to perform such calculations. Initial conditions are determined by a steady-state model that simulates the power-time history of the fuel rod to the point at which the transient begins. The computer codes then attempt to estimate fuel and cladding behavior. In the central region of the fuel rod, the temperature rise due to the substantial zirconium-steam reaction will lead to rapid expansion of the cladding. The general way in which this will be accomplished will be in dicated using the relatively simple model in the earlier FRAP_T21 7,224 pro grams. It is assumed that gas flows from the gas plenum to the expanded region through the fuel-cladding gap. Laminar flow is assumed in the gap. Gas pressures are computed as a function of axial location and time. Ex ternal and internal pressures are used to determine stresses in the cladding.
T H E RMAL A NALYS I S O F PWRS
226
As cladding temperature rises, stresses can reach a level sufficient to cause ballooning at some location. The shape of the localized cladding bulge is taken as that of a rotated sine curve. The equilibrium equation for a thin membrane element in the bulge area can be written as
(2.255)
where
P
=
(Tn,ao = Lc =
differential pressure axial and circumferential stresses, respectively axial and circumferential radii of curvature, respectively cladding thickness.
Significant deformation is not obtained until axial and radial stresses in the cladding exceed the yield Sltress. This criterion may be expressed as o fa ) + 0 fc >
ay / P/ Lc / - f
(2.256)
-
When 1 < 1 for a given axial node, the node will deform. Specification of these deformations is based on the following:
=
1 . Nodal deformations are a function of the nodal instability at that node which is the most unstable (1m maximum 1; ) and will deform the most. 2. The specified displacements must be small enough that adjacent stable nodes are not unrealistically affected. In FRAP-T5,224 deformations are specified by the addition of a finite defor mation to the nodal deformation calculated during the last timestep:
di
i. i, dhi dhi=dhm [ 1 - fm ] i, fill
i , d io
(2.257)
where is the new radial , coordinate of axial node is the old radial coordinate of axial node and is the specified incremental radial dis placement of node Deformations are computed by 0 -1; ) 2
d hm
+ 0.1
(2.258)
where is the maximum displacement to be added to any node, Ii is the instability factor at node and is the maximum instability factor. A value of equal to the cladding thickness was found to produce a rapid con vergence. Although the deformation is not associated with a particular timestep, a limit on the maximum temperature change over a timestep as sures that timesteps remain short during any rapid transient.
d hm
F U E L E L E M E NTS
i.
The effect of adding an incremented deformation to node i is to decrease th e radi us of curvature at Examination of Eq. (2.256) shows this decrease in curvature to increase the stability function fi and, thus, the local stability at node i. An additional effect is that an increase in deformation at node i causes an increase in curvature at node i + 1 and i - 1 . An increase in cur vature decreases the stability function and, hence, local cladding stability. The effect, therefore, of locally deforming a weak spot is to strengthen the weak spot but propagate the weakness into the surrounding material, pos sibly causing additional new instability and further propagation. Careful examination of membrane instabilities on ballooning tubes indicates that initial deformation is quite localized and then proceeds to either rupture or an enlarged stable geometry. In other approaches to rod ballooning problems, the rapid cladding strain rate in the plastic region is used to calculate the cladding deforma tion. In the FRETA program,22 7 the circumferential stresses are computed from thin-shell theory in which the radius of curvature replaces the clad ding radius. The plastic strain f, is then obtained from a relationship be tween stress and strain rate appropriate to the plastic region. In this region, the relationship between stress (J' and strain rate £. can be expressed as (2.259) where
= stress, Pa strain and f = strain rate, s strength coefficient, Pa, exp(8.755 + 8663 / T) for T > 900 K strain hardening coefficient, 0.0279 for T > 850 K m = strain rate sensitivity coefficient, m = 6.47 X 1 0 - 2 + 2.203 X 10 - 4 T for 900 < T < 1090 K and T > 1 255 K.
E= K= n= (J'
K=
n=
-
A strain rate correction is required for 1090 < T < 1255 K. In both the FRETA and early FRAP-T programs, the computed defor mations provide a maximum circumferential strain, which is then com pared to the allowable burst strain at the given temperature. Tests have shown that the burst or rupture strain depends on the rate at which the cladding is being heated. Powers and Meyer27 have examined an extensive set of Zircaloy rod burst strain obtained in an aqueous environment and have developed a tabular relationship between rupture temperature and burst strain for slow and rapid temperature ramps. A shortened version of their tabulation is shown in Table 2.1X. The table also shows the percent of the coolant flow area blocked by the rod burst. It may be observed that the rupture strain increases up to about 800°C, which is about where the tran sition from the ex to Zircaloy phase begins. The strain then drops sharply
�
T H ERMAL A N A L Y S I S O F PWRS
228
TAB L E 2 . 1 X Circumferential Strain a t R u pture o f Zi rcaloy Tubing i n H ot Aqueous Environment*
Rupture Temperature (DC)
Slow Ramp Correlation « 25°C/s) Flow Blockage Burst Strain (%) (%)
Fast Ramp Correlation (>25°C /s) Flow Blockage Burst Strain (%) (%)
600 650 675 700 725 750
10 13 20 45 67 82
6.5 8.4 13.8 33.5 52.5 65.8
10 12 15 20 28 38
6.5 7.5 1 0.0 13.8 20.0 27.5
800 850 875 900 950 1 000
90 82 67 43 25 33
71 .5 65.8 52.5 35.7 1 8.0 24. 1
57 60 57 45 25 35
43.3 46.0 43.3 33.5 1 8.0 25.7
1 050 1 075 1 1 00 1 1 25 1 1 50 1 200
33 25 14 11 10 10
24.1 1 8.0 9.2 7.0 6.5 6.5
77 80 77 39 26 36
61 .6 64.5 61.6 28.5 1 8.3 26.2
*Table derived from data of Powers and Meyer (Ref. 227).
800 1075°C, 950°C.
from to A rise in the burst strain then occurs. A final drop occurs at about which corresponds to the completion of the transition to the phase. In later versions of FRAP-T,2 1 7,228 burst strain has been abandoned as a failure criterion. In the BALON2 deformation subcode,2 1 7 the true stress is used as the failure criterion. True stress, (Jt r ' is defined as the force per unit area using the actual cross-sedtional area at the time of measurement. This differs from engineering stress, (Je ' where the cross sectional area corre sponds to the initial (undeformed) area. The two stresses are related by
�
Ll
Ll
(2.260)
where ftr is the true strain = L / Lactt L is the change in length, and L act is the length at instant of change. The deformation model is used to determine the true stress and strain. The true stress so determined is compared to the true ultimate strength, (J II ' which depends on the strain rate and cladding properties at the given tempetature. According to Hagrman et al.,34 this is given by
(J", �K(lOt-SC �m)'
(2.261)
F U E L E L EM E N TS
- 1 ),
229
where E is the true strain rate (s (T il is the ultimate strength (Pa), and other quantities are as defined previously [see Eq. Calculation o f the deformation considers spatial variations by dividing the ballooning region into a network of axial and circumferential nodes. The model accounts for circumferential temperature gradients, fast or slow hea ting of the cladding and the effects of prior irradiation and cold work. The FRAP-T model considers only a single rod and does not model the effect of neighboring rods. Some programs such as the FRETA226 and the CANSWEL subcode,229 consider the effect of adjacent rods in modifying the shape of the ballooning hot rod. Significant swelling in the region near the spring can exceed the working range of the grid. This can jam the fuel rod against the grid preventing axial motion and inducing fuel rod bowing. The effects of ballooning can, however, be more significant. Interaction between adjacent ballooned rods can effectively block a channel. While such blockage does not have a del eterious effect at high flooding rates (rates in excess of one inch per minute), at lower rates the effect can be deleterious. 23o Current regulations conservatively require that at these flooding rates cooling calculations are based on the assumption that cladding temperatures are determined by steam cooling alone and that the effects of flow blockage are considered. Hall and Duffy231 suggest a procedure for calculating the effect of cladding ballooning on coolant flow. They estimate the flow in the blocked region by assuming that the flow must redistribute in such a manner that pressure drop across the blocked region must be equal to that of the unblocked region in a parallel channel.
(2)
2.6.2.3
(2.259)].
(1) (1994)
Sta inless-Steel Behavior During a L DCA
Stainless (austenitic) steel cladding will react to form hydrogen and metal oxides at high temperatures just as zirconium does. At temperatures above about the stainless-steel oxidation rate also follows a parabolic rate law when steam diffusion is not limiting. 232 If reaction rates are compared on the basis of the mass of oxygen reacted per unit area per unit time, the stainless steel rate is lower than tha t of zirconium until approximately 1400 K. H owever, the rate exceeds that of zirconium at higher temperatures. At about K, the reaction rate constant [see Eq. defined in terms of mg2 O2 / (cm4 s) is nearly times that of zirconium. At K, the rate con stant is only about of that of zirconium. 232 Stainless steel melts and disintegrates ("froths") at about Because of this behavior, the U.s. NRC sets the maximum cladding temperature allowed during a LOCA at the same level as for Zircaloy The heat release in the stainless-steel ! steam reaction is about a factor of lower than that for the zirconium /steam reaction. 232 This would signif icantly reduce the rate of cladding temperature rise during a LOCA. In addition the hydrogen release for unit mass of metal reacted is only two-
1000°C,
1600
20
35% 5.5
(2.250)] 1300 2550°F. (2200°F).
T H E RMAL A N A L Y S I S OF PWRS
230
thirds that of zirconium. Since mechanical properties (strength and ductil ity) of stainless steel are superior to those of Zircaloy at temperatures below less cladding ballooning and deformation during a LOCA would be expected than with Zircaloy-clad fuel.
2200°F, 2 .6.3
Fuel Melting
Although overpower transients due to a rapid reactivity insertion are not now believed to be a significant contributor to severe accident risk in PWRs, such accident scenarios have received considerable attention in the past. Based on experimental observations of actual Zircaloy-clad U02 fuel ele ment behavior during rapid transients in test reactors such as TREAT, 233 it has been concluded that fuel fragmentation occurs when a portion of the fuel becomes molten. The fragmentation is caused by the approximately increase in volume (see Sec. that occurs on U02 melting. The expansion, and possible contact of molten fuel with the cladding, results in catastrophic failure of the cladding. The power burst due to a rapid reactivity insertion is generally com pleted before any significant amount of energy can be transferred from the fuel to the coolant. Hence, the parameter that primarily determines whether the fuel fails is the total energy input per unit mass of fuel. If this energy input is sufficient to cause fuel melting, failure will occur. Experimental data from TREAT233 and similar tests indicate that cata strophic failure will occur at energy inputs of about cal! gm fuel. This observation is the source of the u.s. regulatory requirement that the max imum fuel enthalpy during an anticipated transient without scram (ATWS) be below cal ! g (Ref. This enthalpy limit should provide adequate protection against the loss of coolable geometry due to centerline fuel melt ing. High power levels can be encountered during an ATWS incident and rapid fuel melting is a possibility. In the unlikely event of a LOCA in which the emergency core cooling system fails, fuel melting will also occur. Since the reactor is shut down shortly after the accident is initiated, fuel heat-up is primarily due to decay heat. Not only is there a significant time period before melting begins, but the fuel melting proceeds relatively slowly. The determination of the time at which centerline fuel melting produces fuel failure then requires a ther mal analysis of the melt progression coupled with a mechanical analysis of cladding behavior. It should be recognized that loss of a coolable geometry due to failure of embrittled cladding may come prior to failure due to centerline fuel melting. If the original fuel rod geometry can be assumed to be retained for a brief period, the extent of fuel melting can be estimated using the parabolic profile modification of the lumped parameter approach [Eqs. and Once the central fuel temperature, TFc , reaches the melting tem perature, TI /lf we can assume that the central fuel region will remain at the
10%
2. 1 .1.3)
300
280
(2.231)].
234).
(2.230)
F U E L E L E M E NTS
melting point while the remainder of the fuel melts. The average temper ature of the remaining solid fuel, TI , is then simply II = ( Till
+
TFS ) / 2
(2.262)
if a parabolic profile holds in the un melted region. difference form, on the fuel then becomes ,
q 11 =
A
heat balance, in finite
( mtm- m ) c aa t + [HF at(am)] [ (Tm - TFs ) ] + ( amm ) cl (Tmat- T1 ) I
TI
+
t
R 'F
t
(2.263)
where the revised thermal resistance of the solid fuel is given by R�
and
m f! m am HF r c, r 1
=
[1 - ( rc / r1 )2 - 2 ( rc / rl) 2 In( r / rc)] / (8TIk) 1
total mass and molten mass of fuel per unit length, respectively = increase in molten fuel mass per unit length during period
(a t)
=
=
heat of fusion (energy/ mass) radius of molten fuel and outer pellet radius, respectively,
with other symbols having the meanings assigned in Sec. 2.6.1 .4. Note that, as the melt region advances, both the heat of fusion and heat required to increase the melting mass temperature to Tm [last term of Eq. (2.263)] must be supplied. If we acknowledge that rc and r1 change but make the ap proximation that the effect of the change in density on melting is such that the ratio of the molten fuel radius to the outer solid radius is essentially unaffected by the density change, then ( rc / r 1 )
= (mlm,) 1 /2
(2.264)
Equations (2.262) through (2.264), together with Eqs. (2.231) and (2.233) can now be solved simultaneously for T I , T2, TFS' am, and (1'( / r1 ) if initial conditions are known. The equations are linear if m is taken as having the value at the end of the previous timestep. Programs computing fuel behavior during a LoeA do not generally con sider the extent of fuel melting within the fuel rod . Under most circum stances, the cladding will fail prior to fuel melting. The fuel geometry will then be altered with pellets from failed fuel forming a rubble bed. Progralns examining severe accidents do examine the progression of fuel melting in such beds. Under these conditions, the heat balance integral method of
T H E RMAL A N A L Y S I S
OF
PWR S
6.S.5.5
Goodman23s,236 or the moving boundary approach (see Sec. and Ref. can be used. The problem is greatly complicated by the possible re location of the molten fuel.
234)
References 2. 3. 4. 5. 6. 7 8. 9. 10.
12. 13. 14. 1 5. 16. 17 18. 19. 20. 21 . 22. 23. 24.
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F U E L E L EM E NTS
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86. 87
88. 89. 90. 91. 92. 93. 94. 95. 96. 97
98. 99.
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229. R. W . Bowring, C. A. Cooper, J. Gittus, and T . Haste, " A Code t o Analyse Cladding Deformation in a Loss of Coolant Accident," IAEA Specialists Meetillg 011 Water Reactor Fuel Performance, Blackpool, UK (1980). 230. F. F. Cadek et al., "PWR Fletch-Full Length Emergency Core Cooling Heat Transfer-Final Report," WCAP-7665, Westinghouse Electric Corporation (1 971 ). 231 . P C. Hall and R. B. Duffy, "A Method of Calculating the Effect of Clad Ballooning on Loss-of-Coolant-Accident Temperature Transients," Nucl. Sci. Eng., 58, 1 (1975). 232. D. F. Fletcher, B. Turland, and S. Lawrence, "A Review of Hydrogen Production During Melt/Water Interaction in LWR's," Nucl. Safety, 33, 5 1 4 (1 992). 233. L. R. Baker and R. C. Liimatainen, "Chemical Reactions," Technology of Nliclear Reactor Safety, Vol 2, L. J. Thompson and J. G. Berkerley, Eds., MIT Press, Cambridge, MA (1 973). 234. "Technical Report on Anticipated Transients Without Scram in Water-Cooled Power Re actors," WASH-1 270, U.s. Atomic Energy Commission, Washington, DC (1973). 235. L. R. Goodman, "The Hea t Balance Integral and Its Application to Problems Involving a Change in Phase," Trans. ASME, 80, 335 (1958). 236. M. N. Ozisk, Heat Conduction, Chap. 1 0, John Wiley & Sons, New York (1983).
CHAPTE R TH R E E
H yd rod ynamics
The rate of heat removal from a reactor core and the dynamic forces on the core and internals depend strongly on the flow behavior of the system. In this chapter, the basic flow characteristics of friction, turbulence, void dis tribution, and depressurization are described. Specific applications such as hydrodynamic problems in the reactor vessel and flow instability in a core are then discussed. The material presented here focuses on water systems and assumes an essentially intact reactor core. The larger class of hydrodynamic problems involved in severe accident analysis is not addressed here. Some discussion of severe accident issues is presented in Chapter 6. 3.1 3 . 1 .1
BAS IC F LOW CHARACTERI STICS Transport Equations for Single-Phase Flow
Under normal reactor operating conditions, the flow along a pipe or chan nel is highly turbulent. We may then assume that the fluid velocity is es sentially constant across the pipe cross section. Under these conditions, the conservation equations for adiabatic flow in the axial (z) direction may be written in differential form as av a p ap p - + - + V- = o az az at
for mass conservation and
(3. 1 )
H Y D R O DY NAMICS
243
-avat + V-avaz + g-e -aazp +gF +g-az = O i ( u + V2 ) +a (H + V2 ) V at 2gcf az 2 geJ + � [ aatp (�) + � ( apaz ) + gfe VF+ gfe ddzY ] dY
p
(3.2)
for conservation of momentum and
J
p
p
(3.3)
for the conservation of energy. The symbols are defined as
F= g= ge = H = = = = Y= z = =
frictional head loss per unit length, L/L gravitational acceleration, L/t2 gravitational conversion factor in SI units) = enthalpy, energy/ mass J mechanical equivalent of heat, mechanical energy /thermal energy pressure, force/ L 2 P U internal energy, energy/ mass V axial velocity, L/t vertical elevation, L axial distance, L density, mass/L3 p
(1.0
H
During steady state, the time derivatives vanish. For isothermal flow, is constant and the energy equation becomes equivalent to the momentum equation. The resulting steady-state energy / momentum conservation equa tion may be rewritten as:
ap g + g ay + � V av az '-v-J ge '-v--J ge az ge az '-v--J -pF
-p-
friction
elevation
(3.4 )
acceleration
The total steady-state pressure drop is thus the sum of the friction, elevation and acceleration terms. If we integrate around a closed flow loop, we may write the total pressure drop, .;l P to ta l , as the sum of integrals of the frictional acceleration and elevation terms. .;l P total
=
� P fri ction
+
.;l Pa ccelerati on
+
.;l Pelev
(3.5)
However, in single-phase flow, the integrals of the acceleration and eleva tion changes around a loop are nearly zero (exactly zero under isothermal
T H E RMAL A N A L Y S I S O F PWRS
244
0. 1
0.09
0.08
0.07 0.06 0.05
�
r;.�
.;..
�
or'
�
z
0 � !::!
0.04 0.03
0.025 0.02 0.0 1 5
0.01
0.009
0.008
10'
1 10 "
Fig. 3.1
I I
" 10'
Z:IO'/ "
$ '
" 10'
z ",
R E YNOLDS NUMBER R
=
3 ' . ' ' 10'
--r2\10'1 f
3 '
$ '
\/!J, V i
JI ::I"�0'. 0.000' 01 l I01 �� -
I
.
lr" O. OOO ao lr- O.OOO, 005 ' 1
Moody friction factor chart [from Trans. ASME, 671 (1944)].
conditions). Determination of the total pressure loss is then reduced to the determination of the frictional pressure loss. The pressure loss must, of course, be matched by the pressure rise across the pump. Hence, flow through the loop is fixed at the intersection of the head-flow curve of the pump and the flow-head loss curve of the loop. 3 . 1 .2
Single-Phase Flow Friction
Single-phase flow pressure drop is generally calculated in accordance with
.
where
Ki
(fL) pV2 + I ( P VT)
� Pfriction = De 2gc
2gc
(3.6)
f = skin friction factor due to wall shear in a straight flow channel. Empirical values for f can be obtained from the Moody curve1 in Fig. 3.1. Ki form friction factor due to the i ' th change in the cross section or restriction in the flow ch.annel. The value of K for sudden expansion Ke and for sudden contrauion Kc are given by Kays and London.2 Their values at high Reynolds numbers are reproduced in Fig. 3.2.
H Y D RO D Y NAM ICS
245
v = fluid velocity through the channel Vj = fluid velocity in the smaller channel at area change = equivalent diameter of the straight flow channel, 4 X cross sectional area perimeter.
De
i
I
A nalytical expressions have been developed to describe the friction char acteristics of various flow regimes. For circular channels, the laminar fric tion factor can be calculated
(3.7)
f= 64/Re
where Re is the Reynolds number. The turbulent friction factor in a smooth channel can be calculated from the approximate relationship
(3.8)
C
3.1
where equals 0.184 for isothermal flow in commercially available smooth tubes. The effect of surface roughness is indicated in Fig. in terms of the ratio of the height of the surface projections, E, to pipe diameter, For commercial steel pipe, E can be taken as �0.00015 ft. 3.1 .2.1
D.
Bends, Valves, Fittings, and Restrictions
Le ,
Pressure losses across bends, valves, and fittings are expressed in either terms of a loss coefficient KI, or a length of pipe, having an equivalent pressure drop. Thus, pressure loss � PI across a given component can be calcula ted by y2 P y2 f = � PI = KI
Le 2gc ( De ) p2gc 3, 5
(3.9) (Le i De)
G22 2 ( l - 2uC+u 2C 2 ) 2gc u pC C
(3 10)
The reader is referred to Refs. 4, and for tabulations of KI and for commonly encountered components. Pressure loss across a long restriction in a flow path can be estimated by adding frictional loss to expansion and contraction losses estimate via the K; values of Fig. However, when the restriction is so short (e.g., a per forated plate) that the vena contracta occurs beyond the restriction, then the pressure drop should be obtained from
3.2.
Gu C, L/ D, 3.I. 3.2
� PI =
.
where = mass velocity upstream of restriction, = vena contracta area ratio, and = ratio of restricted flow area to full flow area. Values of the vena contracta area ratio, and the length to pipe diam eter ratio, at which the vena contracta is located are indicated in Table
3.1
Observe in Table that, for Re < Xl, the values of Kc and Kc given in Fig. do not approach 1.0 as the area ratio approaches unity. Todreas
T H E RMAL A N A L YS I S O F PWRS
246
1 .3 1 .2 1 .1 1 .0
� 0.8 "
0.9
� Kc �� 0.6 Turbu lent Re � �. � ggg 0.5 � --... F:::$ -�... � � 1 0 �OO\ � 0.4 - .. - -� � ... - - - �'";� 0.3 ,l� � �� 0.2 - l\F""_"':':=��:;; � �� 0. 1 � �..... r--::: - 0.0 V /� Ke V / -0. 1 Turbu l ent _y V � Re -0.2
�
OJ
"I11III
�
_
..... .-:::0
.:::: � � -:
�-
"'"
1 0 000 � 5 000 � V 3 000 00
-0.3
-0.4
�
-0.5 -0.6 -O J
-0.8 0.0
0. 1
0.2
0.3
0.4
0.5
0.6
OJ
0.8
0.9
1 .0
u
Mi ni mum F l ow Area/Maximum F l ow Area Fig.
3.2
Form friction factors for sudden expansions and contractions [from W.M. Kays and A. L. London, Compact Heat Exchangers, The National Press, Palo Alto, CA (1 955)].
and Kazimi6 point out that this arises because the velocity distributions assumed in the computations are not identical on both sides of the area change. This is the case for entrance and exit area between a large flow area and a restricted flow area (e.g., inlet and exit of reactor fuel assem blies). However, for less drastic geometry changes, the flow distributions will be similar. For large values of (J, the curve for Re = 00 is likely to be a closer approximation than the curve shown for the given Reynolds number.
H Y D R O D Y NAMICS
247
TAB L E 3.1 Vena Contracta Properties (J' =
(Restriction Area) Flow Area
Downstream Location
0 0.3 0.6 0.8
C
(Lj D)
0.617 0.643 0.712 0.813
0.65 0.55 0.3 0.25
The loss coefficients of Fig. 3.2 and Eq. (3. 1 0) are for sharp or sudden area changes and the loss coefficients for more gradual area changes are significantly reduced (see Ref. The loss coefficient for a sudden contrac tion is also reduced by simply rounding or chamfering the entrance. By rounding the entrance with a radius equal to 1 / 10 the diameter of the smaller pipe, the loss coefficient is reduced to 1 / 3 of that given in Fig. 3.2. A 45-deg chamfer at the edge of the entrance to the smaller line reduces the loss coefficient to 1 / 6 of that for a sharp entrance. Rounding or cham fering does not significantly affect enlargement loss coefficients.
4).
3.1 .2.2
Effect of Heating
In a channel where one wall is heated or cooled, the friction factor is changed due to the change of viscosity near the wall. For turbulent flow, Sieder and Tate7 suggested
lllIso =
( )0.14 I-Lwall
I-Lbulk
(3. 1 1 )
where lISO is the friction factor measured under isothermal conditions, and I-L is fluid viscosity. For water flowing at 2000 psia, the effect has been correlated by Esselman et al} LeTourneau et al} and Mendler et al.lO for Re > 3000 as where
111150 = 1 - 0.001 9 il Tf
il Tf = =
h
kb =
Pr/J ' Re/J
=
h
(3.12)
0 0
q" / 0.023 ( kb / De ) (Reb ) . 8 ( Prb ) .4 thermal conductivity of fluid at the bulk temperature Prandtl and Reynolds numbers, respectively, at bulk temperature.
A limiting value of 0.85 has been given for Eq. (3.12) by its authors. Rohsenow and Clark l l examined the Sieder and Tate7 relationship using water flowin2: at oressures of 1500 to 2000 psi. Thev found the exponent
T H E RMAL A N A L Y S I S O F PWRS
248
forpressure Eq. (3.11) was 0.60 for the frictional pressure loss alone, but 0. 1 4 for the loss due to bothgave momentum frictional lo0.85. sses. The Theyrelationship noted that below many of the data points values forandflflso ) ffriction (3. 1 3) ( flSO-friction is recommended. Inflow laminar upflow, heating leadsmaketo anbuoyancy increase effects in the friction factorAswhen the is sufficiently low to significant. thely power to fl o w ratio is increased, frictional l o sses are increased. For ful 1 2 propose the empirical correlation developed laminar(flow, Bishop et al. [ ( ] e r (3.1 4a) [6 ) Re= 1 + 2.97 �to� ) =F' where Gr Grashof number = p2 � g q"/ (kf-L2) q" = heat flux, energy/area k = thermal conductivity, energy/time length T viscosity of liquid, mass/length time � = coefficient of volume expansion = - (dp/ d T) 1 / p T temperature and Bishopotheret al.symbols indicatehave that their previous meaning. For developing flow, (-16f ) Re=l + ( 1 + F'1 ) (3. 1 4b) 0.36 Z - 3 where Z�(LIO) / 50. Ianello et , a l. 1 3 suggest a somewhat more complex ap proach that can also allow for downflow. f-Lwa ll f-Lbulk
0 .6
0. 655
04
=
!.1 =
=
3 . 1 .2.3
Core Pressure Orop
The coresHence, of almostflowalcharacteristics l pressurized water reactors consist of anofarray ofbundles rodreactor bundles. within the core are those theto of which it is composed. Core pressure drop can be considered be composed of three components: 1. pressure loss along the bundle of bare rods 2. pressure loss across the spacer grids or wires 3. pressure loss at core entrance and exit. Entrance exit lossesascanpreviously be evaluated as those due to sudden expan sions andandcontractions, described.
H Y D R O DY NAMICS
249
The pressure loss along bare rods is often predicted by assuming simi larity to flow inside a tube and by using the equivalent diameter concept. In laminar flow, a somewhat better prediction can be obtained by using the so-called "equivalent annulus" approach. In this approach, the equivalent annulus flow area is assumed to be the fluid-dynamic equivalent of the section of annulus extending from the inner radius to the radius of maxiis obtained from: mum velocity, rm ' The ratio of rm to the rod radius, (3.1 5a) for triangular arrays and (3.15b) for square arrays with ( pi D) indicating the ratio of rod pitch to diameter. Todreas and Kazimi 14 show that the product of the Reynolds number and friction factor, I, may then be obtained from
R
j - Re =
2 64(R -3R4 +4R4 InR-1 +4R2 - 1 )3
(3.16)
where = rmlro and the Reynolds number, Re, is based on the equivalent diameter of the flow passage. In turbulent flow,* the assumption of similarity to flow in a tube using the equivalent diameter is also commonly applied. However, there is some evidence that the effect of the pitch-to-diameter ratio ( pi D) is not fully de scribed by the equivalent diameter approach. Deissler and Taylor 1 5 con ducted an early investigation of the validity of this approach for predicting rod bundle pressure drop. By an iterative procedure, they plotted the ve locity profile in square and triangular lattices as shown in Figs. 3.3 and 3.4, respectively. These velocity profiles were calculated from a generalized ve locity profile 1 6 based on Deissler1 7 and Laufer' s 18 data. From these velocity profiles, friction factors and Reynolds numbers can be calculated by inte grating the profiles to obtain bulk (or average) velocities, since both friction factors and Reynolds numbers are based on bulk velocities. Predicted fric tion factors for triangular and square arrays are shown in Figs. 3.5 and 3.6. Deissler and Taylor' s15 predictions are in the vicinity of the line based on *Studies by Haldar, Mohanty, and Gupta [ Heat & Mass Trallsfer 94-Pnlceedings of First ISHMT-ASME Heat and Mass Transfer Conf, ISHMT, Bombay, India (1994)] indicate that there is a gradual transition from laminar to turbulent flow in rod bundles. They find the flow to be fully laminar at Re = 800 and fully turbulent at Re = 104 In the transition region, f= C [ / Re + C2 with C [ and C2 determined by the fully laminar and turbulent values at the endpoints of the transition curve.
T H E RMAL A N A L Y S I S O F PWR S
250
Square a r r a y ;
R e y no l d s n umbe r .
IW 000 ; s pac i ng p a r ame te r .
1 . 1
Fig. 3.3 Predicted velocity distribution between tubes (from Ref. 1 5).
circular tube data. However, there is an effect of pi 0 that would not be observed if the Oe concept strictly held. The experimental data shown have too much scatter to indicate any trend. A number of other investigators have considered the effect of the pi 0 ratio: Miller tested a 37-rod bundle of 3-ft heated length with 0.62S-in.-o.d. rods in a triangular lattice of 1 .46 plO ratio and found the friction factor represented by
et al.1 9
f= 0.296 Re -
et al.9
(3.1 7)
which is 60% higher than that obtained by fitting the Moody curve. Le Tourneau tested rod bundles of square lattice with pi 0 ratios of 1 .1 2 and 1 .20 and of triangular lattice with a pi 0 ratio of 1 .12. These data
H Y D RO DY N A M I C S
/
/ /
T r i an g u l a r a r r ay ;
R e y no l d s number
20 000 ; s pac i ng p a rameter ,
1 . 1
Fig. 3.4 Predicted velocity distribution between tubes (from Ref. 15).
fall within a band between the Moody curve (Fig. 3.1 ) for smooth tubes and a curve 10% below that in a Reynolds number range 3 X 103 to 105 Wantland 20 tested a 1 00-rod bundle of square lattice of 3/ 1 6-in. o.d. and 6-ft length with a pl O of 1 . 106 and 1 02-rod bundle of triangular lattice of 3/ 1 6-in. rod o.d. and 6-ft length with a pi 0 ratio of 1 .1 90. For Prandtl num bers between 3 and 6, he obtained 1.
For square lattices at Re = 10 3 - 104 and Pr = 3 - 6,
f= 1 .76 Re
(3. 1 8)
which is �30% higher than Eq. (3.8) . 2. For triangular lattices at Re = 2 X 1 03 - 1 X 1 04,
+
f = 90.0 Re - 1 0.0082
(3. 1 9)
which is �25% higher than Eq. (3.8). Dingee and Chastain2 1 tested 9-rod bundles in the water of O.5-in. rod o.d., 24-in.-Iong square and triangular lattices with a pi 0 of 1 .12, 1 .20, and
T H E RMAL A N A L Y S I S OF PWR S
��j)
�o_:� ,
0 .0 1 0
t-�
r---...
-- """'" - .......
�
0.00 1
��
r--
----
---
--
- -
I
1
De i ssler's Predi ct ion Ci rc u l ar Tube 0 , 6
1 .12 1 . 20 1 . 27
p/d r-- � r""-- t"-- """ r-- ......
i"'- r--�
roo.
I!.
�
, P I!. t-....� d "': �--""""� -.... r---Kl.. �r:::...., :-_2.0 ......
,...
r--- ....
p- ""'- r- - .... �y
�------ ""'-""'-
'"'"-
1 00 000
1 0 000
r- I-�
�
i"" i""
1 000 000
Re Fig. 3.5 Comparison of predicted and measured friction factors for square lattices (from Ref. 15).
5 20% 15%.
1.27
higher than Deissler and Taylor's Square lattice data are to theoretical prediction.22 The triangular lattice data agree with the same pre diction, but with a scatter of about ± Subsequently, Trupp and Azad23 measured velocity distributions, eddy diffusivities, and friction factors with air flow in triangular array bundles. These data indicated friction factors somewhat higher than Deissler and and 1 05, their Taylor' s predictions. 1 5 For Reynolds numbers between higher than circular tube data. The data at were data at higher than circular tube data. Trupp and Azad 23 were correlated their triangular array data by
104
p/O = 1.5p/0=1.2�27% �17%
f= = 0.287[(2\13/'TT)(p/ of - 1.30J } 104 105 n = 0.368(p/ 1.2 p/ 1.5 p/O C
where
C
Re - n
(3.20)
� Re �
0) - 1 .358
�
0�
The more recent estimates of effects on rod bundle friction in tur bulent flow have tended to rely on computations based on velocity and
253
H Y D RO D Y NAM ICS
� (jJ-g_8� , 00
0 .0 1 0
.......
---
\
P
-- -
h -� r--."" . " ,., ...... l'-..... � ..... I"-t-�
�
-r-...,
--....
f-- - .
r-- ---r-- r--
-r- I"-
-
o 1 .12 , 1 . 20 A 1 . 27
p/d
E ���A d � � �f...-...t � � ���
r""I'" ,...�
-r-. I"-
I
De i ssler's Predi ction C i rc u l a r Tube
- ....
0
.......
.< -.....;;.,.
�
r------ • r- r-
��
to-
I-
r--
1 . 02
-�r- � � 1 .0
"""""" �
0.001
1 00 000 Re
1 0 000
1 000 000
Fig. 3.6 Comparison of predicted and measured friction factors for triangular lattices (from
Ref. 1 5).
eddy diffusivity profiles. Rehme24 developed an approach for calculating rod-bundle friction factors based on a turbulent geometry factor that de pended on ( r Re) for laminar flow in an equivalent annulus. He found that his predictions fit a variety of experimental data for rod bundles and non circular channels. Cheng and Todreas25 and Todreas and Kazimi 1 4 have used Rehme' s method to calculate rod-bundle friction factors as a function of for interior channels and channels adjacent to a wall. They expressed their turbulent flow results as
pi D
f=
C ' Re - O . 1
8
(3.21 )
where Re is based on the De for the channel under cunsideration. Values of C ' are tabulated in Table 3.II for several geometries and values. Computation of via Eq. (3.21 ) and Table 3.II yields values which indicate the same general trend as shown in Figs. 3.5 and 3.6. The predictions of Eq. (3.21 ) are in reasonable agreement with Figs. 3.5 and 3.6 at low Reyn olds numbers (e.g., Re = 20 X 103 ) but they tend to lie above the values of the figures at high Reynolds numbers (e.g., Re > 1 00 X 1 03 ) .
f
pi D
T H E R MA L A N A L Y S I S O F PWR S
254
TABLE 3 . 1 1 Values o f Rod B u ndle Friction Factor Coeffi cient a s a Function o f Geometry
Rod Arrangement Square array Triangular array
Channel Geometry Interior Edge Corner Interior Edge Corner
p/ 0 = 1 .05
p/ D = 1 . 1
0.120 0.129 0.138 0.142 0.129 0.152
0.140 0.147 0.147 0.147 0.148 0.144
C
p/ D = 1 .3
p/ D = 1 .5
0.152 0.152 0.150 0.154 0.152 0.151
0.154 0.153 0.150 0.156 0.153 0.151
Pressure losses across spacer grids or wires are from drag-type pressure losses. These are often calculated using pressure-loss coefficients for sudden contractions and expansions. Berringer and Bishop26 tested the Yankee fuel rod bundle with botij cylindrical ferrule spacers and egg-crate strap grids. The test section bundle was composed of 36 rods of 0.335-in. o.d. in a square lattice with a p/ D of 1 .27 The ferrule is 0.5 in. long with a 0.265-in. o.d. and a 20-mil wall thickness. The strap is 1 1/8 in. high and 1/32 in. thick. Comparison of the ferrule- and strap-pressure drops indicates that pres sure drop across a level of straps was more than four times that across a full-ferruled section. Prediction of the pressure drop across the ferrule or strap using the loss coefficients for sudden contraction and expansion agrees with the data within 1 7% . Friction factors for the bare rod section are � 1 5% higher than the Moody curve in a pipe in the Reynolds number range of 1 0,000 to 30,000. De Stordeu�7 measured the pressure drop characteristics of a variety of spacers and grids. He correlated his results in terms of a drag coefficient, Cs • Pressure drop � Ps across the grid or spacer was given by � Ps = pCs V ; s / 2gcA
(3.22)
where A = unrestricted flow area away from the grid or spacer, Vs = velocity in the spacer region, and s = projected frontal area of the spacer. The drag coefficient used was a function of the Reynolds number for a given spacer or grid type. At high Reynolds number (Re = 1 05 ) strap-type grids showed drag coefficients of � 1 .65. The pressure drop across a crossed circular wire grid was � 10% lower. On the basis of tests of several grids, Rehme28 concluded that the effect of the ratio (s/A ) is more pronounced than indicated by De Stordeur. 27 Rehme concludes that grid pressure drop data are better correlated by � Ps = Cv ( pV � / 2gc )(s/A ) 2
(3.23)
where C v = modified drag coefficient and VB = average bundle fluid veloc ity. Drag coefficient Cv is a function of the average bundle Reynold ' s num-
H Y D R O D Y NAMICS
255
Fig. 3.7 Sketch of typical spacer grid (no mixing vanes) [from Kim e t a/., NIlc/. Techno/., 98, 349 (1993) ] .
ber. Rehme's data indicate that for square arrays, Cv = 9.5 at Re = 104 and Cv = 6.5 at Re = 105 Published information on the pressure drop across mixing vane grids is scarce. Yang report pressure drops across several mixing vane grids in terms of a K factor [see Eq. (3.6) ] . Their values of K varied from approx imately 0.4 to 0.8. Unfortunately, the exact geometries of the grids were not specified. A rough estimate of the pressure drop across such grids may be obtained by using Eq. (3.23) and incorporating the projected area of the mixing vane into the value of s. Figure 3.7 shows a sketch of a typical spacer grid for a square lattice. Although mixing vane grids provide the benefits of substantial flow mix ing, they also substantially increase the pressure drop and hence the up ward hydraulic drag on the fuel assemblies. An appropriate design must ensure that the hydraulic forces do not lift the fuel assemblies. Where the assembly weight may be insufficient to provide the hold-down required, the mechanical arrangement at the upper core plate must provide the ad ditional hold-down needed .
et al.29
T H E RMAL A N A L Y S I S O F PWRS
256
pi
Rehme28 also correlated a wide selection of data on the total pressure losses in wire-wrapped bundles. These pressure losses depend on the D ratio, helical spacer wire pitch H ' and the number of rods. Rehme corre lated rod-bundle' pressure loss data in terms of a modified form of the friction factor, f Pressure loss is given by dP
=
f'
for 4 X l 03 ::s Re ::s 5 X l05 where 51)
/ St
Ve ff
d ill
f'
=
=
=
=
N �e =
( SSbt ) p V �ff L
(3.24)
2 gc De
ratio of the wetted perimeter of rods and wires to the total wetted perimeter effective velocity accounting for swirl flow due to wire D ) 2 ] 2 - 1 6 } 1;2 D )6. 5 [7.6(d'l wraps = (VB
){(pl +
l
/ H' )(pl
mean diameter of wire wraps modified friction factor = (64 N �e ) + 0.0816 modified Reynolds number = Veff De 1-1 .
/ p
/
/
( N ��e )00 1 33
The total pressure drop across a core or reactor vessel is often expressed as dP
p
Ll P
core
pV2 2 gc
= Kcore -
vessel = Kvessel
pV
2 2 gc
(3.25)
where is average density in the core and V is average flow velocity inside the core. For example, K values for the former Yankee Rowe reactor are Kcore = 1 5 ,
Kvessel
and K values for the SELNI reactor are Kcore =
18,
=
34
Kvessel = 52
Lateral flow (normal to the fuel rod axis) occurs in a core where there are significant pressure differences between assemblies. In the lateral direc tion, the core appears as a large tube bank. The pressure drop across bare tube bundles (rod axis perpendicular to flow) is generally computed by an equation of the form: 1 f 'N 2 G max I-1w °0 4 �P= (3.26)
) 2gcp ( I-1b
where
G m a x = mass flux based on minimum flow area N =
number of tube rows in flow direction
H Y D RO D YNAMICS
257
I
f = empirical friction factor ' /-Lb /-Lw = viscosity at bulk and wall temperature, respectively. The value of f depends on the tube arrangement and the Reynolds number. A ccurate values of f were developed by Zukauskas30 for a wide range of Reynold s numbers and are tabulated in heat exchanger design references However, approximate values of for the turbulent region (e.g., Ref. may be obtained from32
I
I
31).
fI
f
0. 1 18 l.08 } Rem-ax0. 1 6 I _ { 0.2S + [(p _-D)/D]
for triangular arrays and f
,
_ -
{0.044
+
0.08( Po/D)+ 1 1 (D(p)} Remax [(p D) / D] .43 . 3
- 0. 15
_
for square arrays. In the foregoing,
Remax = 3 . 1 .3
(3.27)
-
(3.28)
D max / i-Lb G
Simple F low Transients
(3. 1 ), (3.2)
(3.3»).
Fluid behavior during flow transients is governed by the basic conservation equations containing the time derivatives [Eqs. and For incompressible single-phase flow under isothermal conditions, the density is constant and we need to consider only the momentum conservation equa tion. As a simple example, consider the loss-of-flow transient in a single closed horizontal loop of constant cross section. With the foregoing as sumptions, Eq. becomes
(3.2)
dV
- =
/
dt
gc
ap
- - gF p dZ
- -
(3.29)
Since dP dZ is constant throughout the loop, we may equate it to the neg ative of the pressure rise across the pump, ( � P) p , divided by the total length of the loop, We may also replace F in terms of the total frictional loss around the loop. We then have
L.
dV
dt
=
gc ( � Pp ) p
L _2L
Cf V 2
(3.30)
where Cf is the total frictional loss coefficient for loop (assumed to be independent of flow rate). By making the very conservative assumption that the pump inertia is negligible and therefore the pressure change due to the pump is negligible during a loss-of-flow transient,
T H ERMAL A N A L Y S I S O F PWRS
258
dV = dt
_
Cf_ V2 2L
(3.3 1 )
_
Upon integration, w e obtain
V=
Vo (Cf l 2 L) Vo t + 1
(3.32a)
------
where Vo is the loop flow at time zero. This is sometimes rewritten in terms of the loop half time, t,
V=
Vo
tit,
(3.32b)
--
1
+
where t, = 2 1. I Cf Yo , which is the time for loop flow to decrease by factor of 2 in the absence of pump inertia. If, at the opposite extreme, the pump inertia is very much larger than the loop inertia, then the flow during pump coastdown would be given in terms of the tp' the pump half time33
V=
Vo 1 + tit p
(3.33)
where tp is the time required for pump speed to decrease by a factor of 2 during coastdown. The assumption of negligible fluid inertia means tp > t , and hence a much longer coastdown would be observed. The coastdown flow under realistic conditions will lie between the values obtained from Eqs. (3.32) and (3.33) and depend on the ratio of (t, I tJJ A large pump inertia can be obtained by the addition of a flywheel. The increased inertia appreciably extends the coastdown period. An analysis of the loss-of-flow accident under realistic conditions is presented in Sec. 6.3 . 1 . Flow transients may also b e caused b y the sudden opening o r closing of a valve or a rupture of a piping line. In such a case, pressure pulses are propagated through the system at sonic speeds. 34 This situation, commonly referred to as waterhammer, is discussed in Sec. 3.3.5.6. 3 . 1 .4
Pump Behavior
PWR main coolant pumps are all centrifugal pumps. Such pumps have a drooping head flow characteristic as shown in Fig. 3.8. Note that the head versus flow curve is independent of the density of the fluid being pumped. As previously observed, the intersection of the pump performance curve with the flow rate-pressure loss curve of the reactor system (see Fig. 3.8) determines the operating point of the system. In a reactor with symmetric loops, the system curve would be obtained considering the flow through
n
HY D R O DYNAM ICS
259
- Syste m c h aracte r i s t i c M P
" operat i ng
\
p ress ure loss
Operat i ng f l ow rate
Pump c h aracte r i s t i c
1 I I
", I
ayo!. flow Fig. 3 . 8 Pump and system head loss curves.
n)
one piping loop proceeding through an (1 / segment of the core and re actor vessel. The total hydraulic pumping power, required is given by
PI
(3.34)
LlPs
where Q = total volumetric flow at operating point (vol/ time), W = total mass flow rate (mass/ time), and = system pressure drop (force/ area). The actual power required is the hydraulic power divided by the pump efficiency. Since the power required increases as the density decreases, it is obvious why main coolant pumps are always located in the cold leg. The pressure at the entrance to the impeller of a centrifugal pump is appreciably below that at the pump inlet. If this low pressure drops below the vapor pressure of the fluid at the temperature at which it is being pumped, vapor bubbles are produced and the pump is said to cavitate. Cavitation can lead to pump damage and must be prevented. Therefore, all pumps must operate with a net positive suction head (NPSH) as spec value at the ified by the manufacturer. The NPSH corresponds to the suction nozzle less the lowest that exists in the pump. The NPSH requirement is another reason for placing main coolant pumps in the cold leg.
(Pip)
(Pip)
T H ERMAL ANALYS I S OF PWR S
2 60
The head versus flow curve is valid only for the pump speed at which it was determined . Thus, it could not be used during a flow coastdown. Further, during a loss-of-coolant accident (LOCA), the flow through the pump may reverse and, under some conditions, the shaft speed may also reverse. It is therefore necessary to have a complete set of pump characteristics. The behavior of a centrifugal pump may be compactly represented by the so-called "homologous curves." For any given centrifugal pump, the effect of pump speed may be eliminated by plotting ( ;)' P/p) / (n/11r ) 2 versus Q/(n/ll,J
h'
where 11, 111' = actual and rated speed, respectively. When the head is ex pressed in terms of (the ratio of the actual head produced by the pump to the rated head), and flow is expressed in terms of Q' (the ratio of flow to rated flow), the resulting curves of [Q ' / (n/nr )] versus / (n/nr) 2] are the homologous curves. In addition to the region of normal operation, where both Q ' and (11/11,) are positive, three other conditions may be distinquished. Therefore, four quadrants are defined as: Quadrant I II
III
IV
[h'
Pump Behavior Normal pumping Turbine pumping Normal turbine Energy dissipation
Q' = Q/Qr >0 >0 <0 <0
(n /l1,) >0 <0 <0 >0
Figure 3.9(a) illustrates the homologous curves obtained for a typical centrifugal pump. Note that one curve is obtained for quadrants I and IV and a second curve is obtained for quadrants II and III. During a large LOCA due to a hot-leg break, the pump could pass from quadrant I through quadrant IV with reverse flow and positive rota tion. It would sub sequently pass through quadrant III where both flow and speed are re versed. Quadrant II is rarely encountered . Quadrant IV can be avoided by adding a nonreversing rachet on the pump shaft. In addition to the head versus flow relationship, it is also necessary to know the pump torque, T ' as a function of flow. Homologous curves, which hold for all pump speeds, are obtained by defining � as the pump torque/ rated pump torque and plotting � / (11/11r) 2 against Q' / (n/11r) 2 . As may be seen in Fig. 3.9(b), two separate curves are again obtained. Wylie and Streeter34 show that it is possible to replace the four homologous curves by two curves by expressing them as (3.35) and
H Y D RODY NAM ICS
4
�
Enerqy
d 1 s s i � ( it 1 V 21
261
�-----,---
(0' < 0, 17/n, > (;: 3
I
I
Turbine � (l.I' < �. n/I7, < O)
2
N ormal r- pumPin g (0' > 0, n/n, > O)
Tu r b i � e �
pumpln g
(Q' > O, 17/17, < 0)
o
-
�----+-----�
1
-2
-1
o
2
Q' / ( n /
Fig.
3.9(a)
Homologous curves for pump head for typical centrifugal pump.
(n/nr)2 +
(Q' ) 2
(n/nr)]}
(3.36)
4>1
where and 4>2 indicate a functional relationship. Use of Eqs. (3.35) and (3.36) simplify the computer representation of the homologous curves. 3 .2
R EACTOR COMPON ENT HYDRODYNAM I CS
3.2 . 1
Velocity Profiles and Turbulence in Fuel Rod Bundles
3.2.1 .1
Velocity Profiles and Wa ll Shear Stress
Velocity profiles in rod bundle channels have been measured by several investigators. The data of Renksizbulut and Hadaller35 for the central chan-
T H E RMAL A N A L Y S I S OF PWRS
262
4
� E n ergy d i s s i pat i o n
Tu r b i n e
(a' < 0 , n/n, > O)
3
(0' < 0, n/n, < O)
N or m a l pumping
2
(0' > 0, n/n, > O)
Z
o f-------+--/""---
-
1
-2
Tu r b i � e � pumpmg
(0' > 0, n/n, < O)
-2
-1
o
2
Q'/( n / n, )
Fig.
3 .9(b)
Homologous curves for pump torque for typical centrifugal pump.
nel of a six-rod bundle with a square array show velocity contours very similar in shape to the predicted contours shown in Fig. 3.3. Similarly, the measurements of Ouma and Tavoulares36 for central channels in a rod bun dle on a triangular spacing show velocity contours very similar to the pre dicted velocity contours of Fig. 3.4. The velocity profiles depend to some degree on the ( pID) ratio and the profiles are, of course, distorted if the channel includes a portion of a boundary wall. The local wall shear stress varies with azimuthal position. In a square array, the wall shear stress is a minimum at the position of minimum rod gap (azimuthal angle = 8 = 0°) and a maximum where the distance between rods is a maximum (8 = 45°). 35,3 7 The ratio of local to average wall shear stress appears to be independent of Reynolds number in both turbulent and laminar flow. The azimuthal variation appears to be enhanced as the
H Y D RO D Y N AMICS
263
(pID)
ratio is increased. However, the azimuthal variations are generally no more than ± 20% in turbulent flow. Similar azimuthal wall shear variations are seen in triangular lattices. The shear stress is at a minimum at the smallest gap and a maximum in the open flow region.36 Measurements of the variation of the average wall velocity across the mini mUlTI gap region in a square rod array are reported by Hooper?? Rehme,38 Renksizbulut and Hadaller,35 and Kjellstrom. 39 All of these in vestigators found that, when expressed in terms of universal variables, the velocity variation was very close to that seen in a round tube. For (pID) values between 1 .1 5 and 1 .21 7 the data for the turbulent region were close to u = 2.7 In y +
+ 4.0
(3.37)
where u + = u / u* y+ y u * / v u = mean value of axial velocity Tw = wall shear stress at minimum gap = kinematic viscosity = (f.L/
p).
v
Detailed flow measurements within rod bundles have been essentially restricted to single-phase flow (1 993). The effect of the presence of vapor on the detailed velocity and shear stress distribution remains to be determined. 3.2.1 .2
Turbulent Flow Structure
In turbulent flow, the velocity varies chaotically with time in magnitude and direction. The instantaneous values at a point are usually expressed as the sum of a mean value and a fluctuating value. That is u = fi + u '
v = v + v'
(3.38)
w = w + w'
where u, v, w are the instantaneous values of velocity in the y, and x directions, respectively, and u ' v ' w' are the components of fluctuating velocity. For irrotational one-dimensional flow in the direction, v and W are zero. The magnitude of the turbulent velocity fluctuations is generally char acterized in terms of the root mean square (rms) of the fluctuating veloci ties. The turbulent intensity ib is defined as
z,
z
(3.39a)
T H E RMAL A N A L Y S I S OF PWR S
2 64
u*
Alternatively, we may define a turbulent intensity in a particular direction. In that case, the value is often normalized using the frictional velocity, For example,
i X = V(U ' ..)2. / U *
(3.39b)
The fluctuating velocities give rise to momentum and energy transfer between adjacent positions in addition to the transfer by molecular diffu sion. This is quantified by defining an eddy diffusivity, For the asym metric flow in a rod bundle, the value of depends on the direction. Thus, for the direction normal to the wall we define
e.
e
(3.40) and parallel to the wall
(em )x = auU'/wax'
(3.41 )
-
For irratational one-dimensional flow, the effective shear stresses are then given by
T,y= [ �+ p(em )YJe�) Tu= [�+ p(em )x ] e:)
(3.42a) (3.42b)
and the rate of energy transfer within the fluid is
cp p [ a+(eq )y ] �� = q '� c p p [ a + (eq )x ] �� = q '�
where
(eq )z, (eq )x =
(3.43a) (3.43b)
x
eddy diffusivity of heat in the y and directions, respectively ex = thermal diffusivity k/ q '� , q '� = heat fluxes in the and y directions, respectively = specific heat at constant pressure.
cp
Cp p x
eq,
It is commonly assumed that the eddy diffusivity of heat, equals the eddy diffusivity of momentum as determined from experimental velocity fluctuation measurements together with Eqs. (3.40) or (3.41). However, a more exact approach sets
H Y D RO DY N A M I CS
eq em =
/ (Pr)f
265
(3.44)
where Prf = turbulent Prandtl number. Zeggel and Monir4o suggest that, for rod bundles, (Pr}f may be correlated by (Pr)f = 0.98 + 2.87/ (Pr Re1 / 2 ) (3.45) The turbulent intensity in a round tube may be related to the eddy dif fusivity via Prandtl' s mixing length hypothesis. * The mixing length theory states that for one-dimensional flow (3.46) (3.47) where 1 is the Prandtl mixing length. By assuming isotropic turbulence and using Eq. (3.39), we then have (3.48) In the near-wall region, 1 is directly proportional to y, the distance from the wall. The value of 1 then increases more slowly until it reaches about 14% of the radius at the tube center. Considerable effort has been devoted to the determination of the distri bution of eddy diffusivities and turbulent intensities in rod bundles. This has been of particular concern to designers of fast reactors who would like to be able to describe the complete temperature distribution in their high flux, tightly packed, wire-wrapped bundles. This approach has generally been termed the The distributed parameter approach has also been of some interest to the designers of tightly packed, triangular lattice, water-cooled reactors. Based on extensive measurements, Rehme41 has concluded that the wall parallel momentum transport across gaps of tightly or irregularly packed rod bundles exceeds the wall perpendicular transport by up to an order of magnitude. In more open lattices, the wall-parallel and wall-perpendicular diffusivities are more nearly of the same magnitude. Based on the extensive experimental data of Rehme,42 Zeggel and Monir40 have developed an empirical correlation for the radial distribution of the wall-parallel eddy diffusivity in a tightly packed bare-rod bundle with a triangular lattice. In dimensionless form, this is
distributed parameter approach.
ea (r,8)
""The eddy diffusivity may also be obtained via the more sophisticated "K-£ model." Although not generally used in reactor core modeling, the approach has been used in conjunction with "porous media" modeling and it is discussed in Sec. 4.2.2.2.
T H E RMAL ANALYSIS O F PWRS
266
where
eft (r,e) = ee (r,e)/ [u*(e)Ym (e)R] Ym (Re) [ril (e) -R ]I R (e) In[efj+ (ea)]1 -13.8Ym (DeI R
= = rod radius = radial coordinate of line of maximum velocity /11 y = distance from rod surface 0.1 1 8 exp[ + 1 ) 0 . 23 6 + (8) + . 3 = 0.215(De/ R ) 4 + 5. 1 (C 1 / De ) 0. 149 - 6.94 C 1 = azimuthal distance between planes of symmetry in neighboring channels.
3.52 a ]
For the central region of the bundle If the bundle is surrounded by a duct, then in the region near the wall (wall to line of maximum velocity),
(3.49) plD
AR = 2.0,
(3.49)
By use of Eq. together with a wall perpendicular velocity profile, the axial momentum balance equation can be solved. Note that Eq. holds strictly only for a ratio of 1 .10. However, it should be a reasonable approximation of the behavior in the tight-lattice, triangular pitch cores with ratios close to 1 .1 . In an actual reactor core, behavior is complicated by the presence of spacer and mixing vane grids or wire wraps. The vapor produced by the boiling in the hottest channels of the core also makes the behavior substan tially more complicated than in a bare rod bundle with single-phase flow. Because of these complications, most current analyses of PWR be havior are based on the or approach. Here, the properties of a single subchannel (region between four adjacent rods in a square array, region between three adjacent rods in a triangular array) are assumed to be uniform. With this approach, the information needed is that required to determine the momentum and mass transfer between subchan nels. Therefore, the turbulent mixing of greatest importance is the wall par allel mixing at the minimum gap between rods. Rehme' s42 data indicate that the turbulent intensity in the minimum gap region is very much higher than in the central region of the channel. Axial and azimuthal turbulent interstices have a similar order of magnitude, but the azimuthal turbulent intensities are about higher than the axial val-
pi D
(1993) lumped parameter subchannel
30%
H Y D RO D Y NAM ICS
267
80%
of the gap appear to ues. The turbulent intensities across the central fairly constant. be Perhaps the most important conclusion regarding the turbulent intensi ties in the rod gap region is that they are above those for round tubes. The difference is greatest when comparing the central region of the gap and the central region of a round tube where they can differ by a factor of 2. Hooper37 and Hooper and Rehme43 also show that the axial and azimuthal turbulent intensities in the gap region increase significantly as the gap size is decreased. Further, compared to triangular rod bundles with similar p / D ratios, the square bundles show higher levels of turbulence in the gap region. Since eddy diffusivity varies directly with turbulent intensity [see Eq. the rate of turbulent momentum and energy exchange per unit area will increase as the gap size is decreased. In addition, for bare rod bundles, the exchange rate per unit area will be higher in a square array than in a triangular array with the same p / D ratio.
(3.48)],
3 . 2 . 1 .3
Effect of Mixing Vanes and Two-Phase Flow
The square array rod bundles of vessel-type PWRs often contain mixing vanes placed on the upper edge of the grid spacers.* The vanes are pro duced by bending tabs of metal which extend above the grid. The entire tab may be bent in one direction or the tab may be split and the halves bent in opposite directions. A typical mixing vane arrangement is shown in Fig. Note that the vane angle, a', represents the angle of the blade relative to the flow direction. Although most of the information on mixing vane performance is pro prietary, some data have appeared in the open literature. The laser-Doppler velocimeter measurements of Shen et provide a picture of the action of the split vanes illustrated in Fig. The lateral velocity profile at two distances downstream of the vane is shown in Fig. At a short distance downstream of the vanes, flow enters from two of the adjacent channels and exits to the remaining two. In addition, a vortex is created in the center of the subchannel. Similar behavior is seen further downstream but the direction of the flows from the adjacent channels have reversed and the magnitude of the flows is substantially reduced. The cross-flows produced by the vanes lead to substantial mixing of the coolant in the bundle. This mixing leads to a considerably more uniform enthalpy rise across the bundle.
3.10.
3. 1aZ.0. 44
3.1 1.
*Some later designs achieve improved interchannel mixing without mixing vanes. In these designs, each grid strap is composed of two thin segments. The segments are attached to each other in a way that produces small channels between the segments in addition to the main channels within the spacer grid. The flow through these small channels, which are tilted at an angle to the axis of the fuel rod, causes the desired mixing improvement. Performance data on these grids remains ( 1 995) proprietary.
T H E RMAL ANALYSIS OF PWRS
2 68
A
1
L A
Fig.
3.1 0
2
A
3
��
________ __ �________
A
Arrangement of mixing blades on the grid spacer. I, grid spacer; 2, ripped-open blades; 3, central subchannel [from Ref. 44J .
H Y D RO DY N AM I CS
(a)
D
ill"
I I
I
I
I
I
I
I
269
\J (b) :I � C
Fig.
3.1 1
Lateral velocity profile for
a = 25°:
(a) L! De 6, (b) L / De 25 [from Ref. 44]. =
=
et
Yang a1. 45 measured total mixing downstream of several grids. The results of this study also showed a high rate of mixing immediately down stream of the grid. The mixing rate decayed rapidly and then remained nearly constant at L ID ratios above 1 0. Yang a1. 45 observed that the de gree of mixing depended significantly on the grid design. The grid having a pressure loss coefficient substantially above the others tested showed the highest mixing rates. However, there was no obvious relationship between loss coefficient and mixing rates among the other grids. However, Shen al.44 found that the degree of mixing depended on the vane angle (a ' in Fig. 3. 1 1 ). Mixing and pressure drops increased as a' increased, but, at large values of a ' , the mixing rates became nonuniform across the subchannel. After taking this nonuniformity into account, they recommend a' = deg as optimum. In addition to providing gross interchannel mixing, mixing vane grids also somewhat increase the turbulent intensity immediately downstream of the grids. 41 The axial turbulent intensity was about higher than bare rod values immediately downstream of the grid. However, after an L I D o� about the turbulent intensity differed little from that seen in a bare bundle of the same pi D. This result indicates that the major effect of a vane is the gross mixing it produces. The turbulent intensity is also affected by the presence of vapor bubbles. The measurement of the detailed distribution of eddy diffusivity and tur bulent intensities in rod bundles under two-phase flow conditions is quite difficult. The effects need to be inferred from limited data on pipe flow and on overall mixing coefficients. Michiyoshi and Serizawa46 examined two phase turbulence during bubbly flow in round tubes using dual-sensor hot
et
et
25
10,
60%
T H E RMAL ANALYSIS OF PWRS
2 70
film anemometry. They found the turbulence to be highly promoted by the presence of bubbles. Their study, which was limited to the low void region, showed that eddy diffusivities in the central region of the tube could be more than double the single-phase values. Although detailed measurements of the eddy diffusivity distribution in rod bundles are not (1993) available in the presence of vapor, data on average rates of turbulent mixing with two-phase flow4 7-5o confirm the significant effect of the presence of vapor. In view of the difficulties in determining local values of eddy diffusivities when mixing vanes and two-phase flow are present, most computations are based on the use of empirical correlations of gap average values. If e, an average value of the transverse (wall-parallel) eddy diffusivity in the gap region is known, the value of w' the turbulent transverse fluctu ating flow rate between adjacent subchannels per unit length of rod (mass/ hL), may be obtained from w ' = pe (l/De)
0
(3.5 )
where p = fluid density, mass /volume e = average eddy diffusivity, area/ time 1 = Prand tl mixing length, L De = equivalent diameter of channel, L . Since the ratio (1/ De) does not change very much for reactor channels, it is common to combine this ral t io with e and define an average mixing coefficient, £, such that w ' = p£
(3.51 )
�0.06
where £ and e leave the same units. For a PWR assembly without mixing (lb / s ft) is a typical value for w ' The various empirical cor vanes, relations that have been used for w ' and £, and which combine the effects of turbulent mixing and gross mixing due to mixing vanes or wire wraps, are described in Sec. 4.2.1 .3. 3 . 2 .2
F l o w in Plenums
Flow streams or vortices in the lower plenum can affect flow distribution at the core inlet. The downward jet formed by the flow coming down be tween the thermal shield and vessel wall causes a low-pressure region at the edge of the core inlet, which will reduce flow in the outer assemblies. Pressure at the center of a vortex with a horizontal plane of rotation is usually lower than the surrounding pressure; thus, the flow rate to any fuel assembly located above a vortex is reduced.
H Y D R O D Y NAMICS
The most effective way to even off the inlet flow is to increase flow resis tance at the bottom of the core. According to Prandtl's equation for velocity distribution improvement,s 1 Ll Vafter screen Aavg
where K
=
1
K+1
Ll Vbefore screen Vavg
Ll Pscreen
pV�vg / 2gc
(3.52)
(3.53)
A similar relationship can be applied to the core inlet. For example, consider the situation where the Euler number at the core inlet is Eu
=
Ll Pcore In1 et � 10 pVTet / 2gc .
(3.54)
Under the most adverse conditions, K is then equal to 1 0 . To obtain the pressure drop at the inlet of the hot channel, we add the stagnation pressure of the jet to the average inlet pressure drop. This leads to (Ll P of maximum flow channel! Ll P of average channel) = 11 / 1 0. Since il P is proportional to V2, the ratio of the maximum inlet flow rate to the average inlet flow rate is (1 1 / 10)1/2 Several analytical procedures based on potential flow theory have been used for predicting the velocity distribution in the lower plenum. Yeh and Tongs2 used conformal mapping to solve a two-dimensional approximation of the flow field. Yehs3 obtained a solution for the potential field in the three-dimensional case. He divided the field into several regions (down comer, lower plenum, and cylindrical region above bottom of downcomer) and solved the differential equations for each region by the separation of variables technique. Figure shows the results of a typical calculation. This figure shows the benefit of a downcomer skirt that extends an appre ciable distance below the core. While the flow is clearly not evenly distrib uted at the edge of the downcomer, distribution becomes more uniform as one moves upward. Although potential flow theory can be considered useful in giving a qual itative view of behavior and an approximate estimate of velocities, analyt ical methods must simplify the geometry and ignore the many structural members in the lower plenum. Of course, any vorticity introduced cannot be considered. In view of the limitations of previously available analytical techniques, most reactor designers determined the velocity distribution in the lower plenum from experimental studies using scale models. Hetsronis4 describes hydraulically testing a model of a PWR vessel and core. Of the four dimensionless groups involved, geometry, relative rough ness, Reynolds number, and Euler number, [.� P (2gc ) / pV 2 ], he concludes
3. 1 2
T H E RMAL A N A L Y S I S OF PWRS
rjro
0.9 0.8 0.7 0.6 0.5 0.4 0 . 3 0 . 2 0. 1
0
I
0.8 0.7 0.6
� �
N
0.5 0.4 0.3 0 .2 0. 1
o
(90° )
Fig.
3.1 2
t I l 1 I 1 1 1 1 1 I ,, II I !
'I '
I I 1 1 1 1 1 1 I ' r t
Typical velocity distribution in lower plenum. The arrows indicate direction and relative magnitude of fluid velocities [from Ref. 53, courtesy Nucl. Eng. Des.].
that geometry and Euler number are most important. His tests used a � scale model that preserved geometry. Care was taken to keep the Euler number similar in the prototype and model. The � scale factor was also adjusted for relative roughness in small flow passages, but it was ignored in the large passages. The test results obtained a significantly improved flow distribution when a straight downcomer skirt extended below the core. A tapered skirt did not lead to satisfactory flow distribution. Flow distri bution was also found to depend on the geometry of the lower plenum. Application of the experimentally determined velocity distribution to an actual core design was somewhat of a problem. Under operating conditions where high power regions generate appreciable quantities of steam, the inlet flow redistributes to maintain a nearly constant pressure at the core
H Y D R O D Y NAMICS
outlet. Khan55 suggested that the measured velocity distribution at the core inlet be used to determine an inlet pressure distribution. Further, he sug gested that the inlet pressure distribution be assumed to hold under op erating conditions. An alternative procedure would be to use inlet flow distribution to determine a set of flow resistances between the downcomer and various core assemblies. These flow resistances then might be assumed to be the same under operating conditions. The degree to which the coolant entering the reactor vessel through the several loops is mixed in the lower plenum of the vessel is of concern under asymmetric flow conditions. Such conditions can develop between reactor cold-leg loops due to instrumentation and control uncertainties, pump flow uncertainties, or variations in the secondary side flow in a steam generator. If such asymmetries lead to asymmetry in the temperature of the coolant entering the core, core power distribution and burn up could be altered. More severe asymmetries are possible during accidents where such things as pump failure, a stuck-open relief valve on a steam generator, or steam line break can significantly alter cold-leg temperature. Mixing tests, conducted on a large PWR having an external system with two steam generators and four reactor pumps, showed very incomplete mixing during normal (four-pump) operation.56 In the lower plenum, only about of the flow was mixed completely and another was partially mixed. Greater mixing was attained with asymmetric operation but the mixing was still far from complete. Little mixing was seen in the core but significant mixing occurred in the upper plenum. Under normal operation, mixing in the upper plenum accounted for to of the total mixing. It would appear that the common accident analysis assumption of a uni form core inlet temperature may not always be justified. If an allowance for minimal mixing in the lower plenum is made in the asymmetric case, an allowance for the mixing in the upper plenum should be included in evaluating the temperature of the coolant entering the various loops. Upper plenum behavior can significantly affect a number of accident sequences. In reactor accidents that involve substantially reduced flows through the reactor vessel, the possibility of thermal stratification in the upper plenum should not be overlooked. During the reflooding phase of a LOCA, the control-rod guide tubes in the upper plenum can cause signif icant deentrainment from the droplet laden steam leaving the core (see Sec. If a severe accident involving core damage were to occur, the upper plenum would play an important role in determining molten fuel transport and sweepout as well as debris collection. The advances of computational fluid mechanics have now considerably changed the situation for designers of future reactors. Large computer pro grams, based on the so-called "porous media" concept, are now available for treating large scale flow distribution problems. Such programs use finite difference approximations of the three-dimensional conservation equations
15%
25%
60 75%
3.5.2).
T H E RMAL ANALYS I S OF PWR S
and include a model for turbulent energy dissipation. Boundaries are gen erally described by local isotropic porosity factors and fluid-solid source terms. The effect of structural members in the plenum can thus be included. In addition, it is possible to model both a symmetric section of the plenum and the reactor core. It is feasible to model the reactor core on an assembly by-assembly basis with the appropriate levels of heat addition. After ad justing the model so that agreement with experiment is obtained for isothermal conditions, the model may be used with appropriate heat ad dition to establish core inlet pressures or flows at design conditions. Porous media computations should also be able to determine the degree of mixing in the upper and lower plenums. The porous media approach is discussed in detail in Sec. 4.2.2 and in Refs. through of Chapter 4.
79
3.2.3
81
Flow-Induced Vibration
It is generally agreed that flow-induced vibration of nuclear reactor com ponents is due to one of four mechanisms, namely: 0) fluid elastic insta bility, (2) periodic vortex shedding, turbulence-induced excitation, and acoustic resonance. The relative importance of the various vibration mechanisms depends on the particular component and flow situation. Fluid-elastic instabilities occur when there is a coupling between fluid induced dynamic forces and structural motion. Such instabilities occur when the velocity is high enough that the energy imparted by the fluid force exceeds the energy dissipated by damping. Periodic vortex shedding occurs downstream of a structure placed in a cross-flow. If the shedding frequency coincides with a natural frequency of the structure, damaging resonance may occur. Turbulence-induced excitation produces random pressure fluctuations, which cause vibration. Turbulence excitation is most important in axial flow. Acoustic noise is present in all systems. It is a vibration problem only if the frequency of the noise is close to one of the natural frequencies of the structure.
(4)
3.2.3.1
(3)
Hydra ulic Vibration o f Core Structure
When a high-speed flow passes through a gradually narrowing passage, the pressure head of the stream is converted into a velocity head, which creates a suction force on the wall. If the wall is movable or flexible, the passage is automatically reduced to zero and flow is stopped. As soon as flow becomes stagnant, the stream pressure increases to its maximum value, pushing the wall back, thus providing a wide passage again. These suction and pushing forces can act periodically and, thus, vibrate the core structure. This fluid-elastic instability mechanism appears to govern the vibration of parallel fuel pIa tes.
H Y D R O DYNAMICS
275
Parellel fuel plate vibration has been analyzed by Miller,57 Scavuzzo,58 and Remick.59 Miller applied a method of "neutral equilibrium" where the pressure and plate restoring forces were balanced. He was then able to determine a critical velocity, V, above which significant vibration will occur. He expresses this vibration-inducing velocity for fuel plate assemblies with long edges attached to side plates as
(3.55)
where
= b d = E= 'Y l =
dL
plate thickness = plate width flow channel depth Young' s modulus of elasticity Poisson's ratio.
Miller' s results are frequently applied, and Wambsganns,6o who extended the work to include second-order terms, concluded that critical flow veloc ities could be from to times those derived by Miller. Miller and Kennison6 1 analyzed the flow-induced vibration of a blade suspended in a flow channel, which widens at a point downstream of the leading edge-a condition similar to that observed by a control-rod blade. They concluded that there are three possible motions: (1) decaying vibra tions, divergent vibrations, and static deflection against the channel wall. The motion observed is dependent on insertion length and flow rate. For short insertion lengths, there is a critical velocity above which divergent oscillations are obtained. They also conclude that a critical insertion length exists above which no oscillations can be sustained, but where deflection against the channel wall is possible. Wambsganss62 concludes that the anal ysis explains experimental observations qualitatively, not quantitatively. Acoustically induced vibrations can be significant for reactor internals (core barrel and thermal shield). Acoustic pulsations generated by coolant pumps can cause large responses in shell structures. This is most serious when several coolant pumps are present. The minute differences in impeller rotation speeds means that at some point in time the acoustic pressure generated by these pumps will reinforce each other. This reinforcement is known as "multipump beating" and has been observed in operating reactor systems.63 The frequencies at which acoustic oscillations may be expected correspond to the pump shaft rotation speed, the vane-passing frequency or their higher harmonics. Serious damage can occur if the natural fre quency of the structure coincides with any of the foregoing. If the natural frequency of the structure is substantially above the fre quency of the expected oscillation, vibration problems will not be signifi cant. The response of complicated structures to an arbitrary dynamic
0.63 0.85
(2)
(3)
T H E RMAL A N A L Y S I S OF PWR S
2 76
excitation is difficult to predict. Bohm64 developed an analytical model that approximates the internals as an interconnected beam-type structure and uses a transfer-matrix approach for determining natural frequencies. Bow ers and Horvay65 adopted a similar approach for estimating vibration fre quencies of the core barrel and thermal shield. They noted that the narrow gap between the shell and vessel was responsible for a substantial increase in the virtual mass of the shell. They expressed their results in terms of a nondimensional frequency parameter. The actual frequency of vibration, w, is given by
v,
(3.56)
where
L= = ms + mh = = = M= v
m ms , mh E
I
length of the shell
mass per unit length of shell and entrained water, respectively Young's modulus moment of inertia of shell attached mass (tubesheet, part of fuel, etc.).
L M), 3.4 4.2, 6.3 7.2, 9.4
They found to be a function of the weight ratio ( m / support stiffness, and bottom restraint. Typical values for v were in the range of to for the fundamental mode; to to and to 1 0.2 for the second, third, and fourth harmonics, respectively. Actual operating experience indicates that the natural frequencies of the core barrel and thermal shields may not be significantly above the fre quency of the exciting pulses. Therefore, in addition to making certain that the natural frequency of the structure does not closely correspond to one of the acoustic frequencies expected from pump noise, it is necessary to show that the actual response of the system is not excessive. This compu tation usually involves63 determination of variation in the spatial acous tic pressure distribution as a function of time (see Ref. an analysis of the transient dynamic response of the structure to determine that struc tural response is not beyond the limit during pump startup or coastdown, and (3) a steady-state vibration analysis to assure that structural response is not excessive during normal operation. Procedures for analyzing the responses of core barrel and shield struc tures to specific pressure loadings have been developed by several inves tigators. Penzes67 shows how a known pulsating pressure at the inlet to downcomer region can be used to determine the pulsating pressure distri bution on the cylindrical core barrel or thermal shield. Penzes introduces the concept of a time-dependent body force (time-dependent additional thickness) in the governing differential equation. With this conceptual sub-
(1)
66), (2)
1.5 2.2
H Y D RO D Y N A M I CS
stitution for actual loading, the time-dependent mixed boundary value problem can be represented by a vibration problem with homogeneous boundary conditions. Bowers and Horvay68 applied Penzes' approach to predicting internals response, but obtained vibration amplitudes consider ably smaller than those predicted by Penzes. The most sophisticated approach devised is that of Au-Yang.69 He ob serves that flow-induced vibration is a stochastic process. Since stress and strain within a structure can be obtained from displacement distribution, Au-Yang develops expressions for predicting rms displacement and dis placement power spectral density as a function of location. The prediction requires a knowledge of the natural frequency and mode shapes of the structure, virtual mass, pressuring loading function, and damping ratios. The loading function and damping ratios must be developed on the basis of experimental information (Penzes ' analysis could be used to determine loading function if pressure fluctuations are known). In-fluid frequencies and modal shapes can be predicted by currently available finite element computer programs providing the virtual mass is supplied to the program. The virtual masses, in-fluid natural frequencies, mode shapes, and experi mentally-based loading functions and damping ratios are used to compute displacement power spectral density in terms of Powell' s acceptances?O The complexity of most reactor coolant systems makes it difficult to pre dict the frequency and magnitude of hydraulic forcing functions; therefore, scale models of system components are often tested under simulated op erating conditions.* 3.2.3.2
Hydra ulic Vibra tion of Fuel Rods
A cylinder exposed to axial flow will exhibit fluid-elastic instability if sub jected to a sufficiently high flow. The critical velocity, Ve , above which fluid elastic instability will occur with a cylinder supported at both ends in an unrestricted medium63 is given by V, = (1' / L )
(:�)
(3.57)
where L = unsupported rod length, p = fluid density, A = cross-sectional area of rod, and other symbols have their previous meaning. This critical velocity is normally considerably above the values encountered in reactor conditions. *Au-Yang and Jordan [NIlc/. Ellg. Des . , 58, 1 1 3 ( 1 980) ] obtained forcing function data for the barrel and shield structure of a 1 / 6 scale model of a commercial PWR. After expressing their observations in terms of dimensionless parameters, they found that the nondimensional re sults also represented the observed behavior in a geometrically similar full-size reactor. Forc ing function data for a 1 /9 scale model of an advanced reactor with larger diameter barrel and shield were obtained by Au-Yang et al. [NIlc/. Ellg. Des ., 157, 93 ( 1 995) J.
T H E RMAL A N A L Y S I S O F PWR S
278
At reactor conditions, the hydraulic vibration of fuel rods with their axes parallel to the flow is due to turbulence excitation. This problem has been studied by Burgreen et Quinn,72 Sogreah/3 Pavlica and Marshall/4 6 Paidoussis/5 and Morris? The earliest correlation is that due to Burgreen, who found that single rods vibrate at their natural frequency (reduced slightly by viscous damping in water) and that amplitudes of the vibrations are given by 1 .3 0. p, V p , V L4 5
al./1
(�) =(O.83X10 -1 O)c ( 2) ( 2 ) o
(3.58)
EI
/-L W I l
where 8
maximum rod deflection o rod diameter C for pin-ended rod, 1 for rigidly held rod, and for one end rigidly held, the other pin ended 4 W , l = the natural frequency of vibration (Hz) = VEI/(M 'L ) water density PI /-L = water viscosity E = Young ' s modulus of elasticity I = moment of inertia V = axial velocity M' rod mass per unit length L = rod length. =
= =5 =
2.08
=
Paidoussis75, 77 extended the work of Burgreen et for deflection amplitudes is
al.7'
His later correlation
(3.59) where
= = u= = =
(X l
B
V Oe
K
=
first-mode beam eigenvalue M + m / M, M = rod mass, m fluid mass (M/ EI) ll2 (VL) axial velocity hydraulic diameter of the flow channel constant that is 1 for quiet flow and for realistic disturbance levels.
=
5
H Y D R O DYNAM ICS
2 79
This correlation shows much less dependence on fluid viscosity than Pai doussis ' original proposal. It brings it into close agreement with the data of Quinn72 and Pavlica and Marshall,74 which show almost no fluid tem perature effect. Paidoussis77 also examined the vibration of 1 9-element wire-wrapped bundles and found vibration amplitudes differed substantially from those of Eq. (3.58) and that they did not increase significantly with flow. Kinsef8 studied the vibration characteristics of 21 7-pin spiral wire-wrapped fuel assemblies in water. He observed very low water pin displacements, gen erally 5 mm or less. The O.23-in.-diameter pins showed distinct resonances at low flows, but broad-band Gaussian distributions at high flows. Pavlica and Marshalf4 examined the vibration of rod bundles with spring clip spacers. In correlating the data, they used the stiffness (El) of the bundle rather than that of the individual rods. Since intermediate rod to-rod connectors are not rigid, they used experimental values of ( El ) rather than an estimate of single-element parameters. With this change, they found their room temperature data agreed with the original Paidoussis correla tion. The discrepancy between their 1 50°F data and correlation appears to be remedied by the reduced viscosity effect in the revised correlation. Gorman79,80 investigated the effect of void fraction on single-pin vibra tion in both an annulus and a seven-rod bundle. Pin dimensions and bundle arrangement were similar to those found in CANDU designs. Gorman found that rms pin displacement increased substantially under two-phase conditions. The maximum rms displacement occurred at simulated qualities in the range of 12 to 20% . At a constant mass flow, the maximum rms displacement under two-phase conditions in a rod bundle could be up to eight times that observed in single-phase flow. Paidoussis81 recommends that the following conditions be adhered to in the design of reactor fuel assemblies: (1) The axial velocity be below the critical velocity for fluid-elastic instability (generally not a problem). (2) The frequency of flow perturbations be at least 25% above or below (2wn ) where Wn is the natural frequency of the fuel rod. (3) The amplitude of the tur bulence-induced vibrations be such that adjacent fuel rods will not touch. In actual practice, conservatism requires the maximum vibration ampli tude to be considerably smaller than would be allowed by Paidoussis' third criterion. Vibration amplitudes of 0.2 to 0.25 mm are common design cri teria.82 To obtain such low vibration amplitudes, a considerable force must be maintained by the spacer-grid springs. Further, if fretting wear at the support is to be prevented, the sinusoidally varying reactions at the support should be less than the spring preload. Preumont82 suggests that these forces may be computed by using a finite element model of the fuel rod with distributed masses and spring forces at each of the support points. He assumes that damping is negligible and that the rod vibration is in its fun damental mode with the maximum deflection at the design value. He then
T H E RMAL ANALYSIS O F PWRS
280
determines the forces at the support locations required to bring the rod displacements at these locations to zero. For a typical fuel rod in a 14 X 1 4 PWR assembly, h e computes that spacer-grid forces i n excess o f 1 N are required at the end supports 0 .5 N at top) but considerably lower forces are needed in the center region of the rod. Note that, due to creep, grid springs relax under neutron irradiation. After a fast fluence of 2.5 X 1 021 n/ cm2, the residual spring force on an Inconel spring is only about 1 / 7 of the original force.82 Thus, the spring preload at the beginning of life needs to be substantially higher than that calculated in order to provide the needed preload at the end of life. How ever, the highest forces are needed at the assembly top and bottom where fluxes are lower. At a fast fluence of �0.85 X 1 02 n / cm2, the residual force on an Inconel spring has only dropped to 1 /3 of its original value.
1
3 .2 . 3 . 3
Vibra tion of Heat Exchanger Tubing a n d In-Core Instrumentation Thimbles
Vibration of heat exchanger tubing can be a significant cause of tube failure. Such tube failures may be due to either fatigue or impact-fretting wear. Vibration problems are most severe in those regions of the heat ex changer where the tubes are subject to appreciable cross-flow. This occurs in 0 ) the inlet region of the heat exchanger where there is a flow of liquid from the entrance ports across the lower portion of the tube bundle, and (2) in the U-bend region of a U-tube heat exchanger where the two phase mixture flows across the bundle. Vibration in cross-flow may be due to either fluid-elastic instability or periodic vortex shedding. With an isolated cylinder, periodic vortex shed ding is a well-understood phenomenon. The shedding frequency is repre sented by the Strouhal number, St,
=
(3.60)
=
where II' vortex shedding frequency, L characteristic dimension (D for cylinder), and V = stream velocity. When the Reynolds number is in the range of 3.6 X 106 to 1 X 107, the vortex is in the transcritical range. The drag remains constant and the Strouhal number remains at 1 .2 as the velocity is changed.83 Behavior in tube bundles is not as well understood. However, the problem appears to be of lesser importance. While difficulties can arise if the shedding frequency corresponds to the natural frequency of the vi brating tube and resonance occurs, fluid-elastic instability appears to be the main problem. Just as in axial flow, fluid-elastic instability in cross-flow is avoided if the cross-flow velocity is kept below a critical value. Pettigrew a1.84 rec ommend that �
et
(3.61 )
281
H Y D R O DY N A M I CS
where D Vr V'Xl p
=
rod diameter Voc p/ (p - D) superficial fluid velocity tube pitch mass per unit length (includes hydrodynamic mass) total damping ratio natural frequency of vibrating tube (Hz).
= =' m� == T= et n = 0.5. (2TI�m pD2 ) 2, 4.9 n ==0.4.008 �s. et (N� 1 ) e��O )( p�2 )( �D 0 6 (3.62) LlL = L N= et Wn
=
/ below 2, Pettigrew al.84 recommend that K For values of and For values above they suggest that taking K = and will provide a lower bound for the critical velocity. � The total damping ratio is the sum of the viscous damping, ," and the support damping Pettigrew a1.84 recommend �, =
where number of spans, support thickness, = characteristic span length (average of three longest spans), and other symbols are as previously defined. The viscous damping in a single-phase liquid may be obtained from the theory of Chen a1.85 but a simplified version due to Rogers et al.86 is valid for the usual range of heat exchanger parameters (3.63)
v=
=
where kinematic viscosity and Dc effective diameter of surrounding tubes. Viscous damping in a two-phase fluid is considerably more complicated than the single-phase damping and no simple equation is available for its calculation. The relative damping of a two-phase air-water mixture has 84 At a void fraction of been measured for a p / D ratio of the viscous damping has been increased by a factor of about 3.5. The maximum damp ing was reached at a void fraction of about Damping then declines but there was considerable scatter in the measurements. Two-phase effects should be considered in flow situations such as in the V-bend region of the steam generator. However, use of Eq. (3.63) would give a conservative value for the critical velocity. To determine whether the critical velocity defined by Eg. (3.61 ) is actu ally reached in a particular heat exchanger, an analysis of the flow field within the tube bundle is required. This can be accomplished by means of computer programs using the porous media approach (see Sec. The THIRST program87 is an example of such a program developed specifically for steam generator design.
1.47
0.6.
0.4,
4.2.2).
282
T H ERMAL ANALYSIS O F PWR S
Vibration of in-core instrumentation thimbles must also be considered. In-core instrumentation is brought to the core via tubes or thimbles that penetrate the lower head of the reactor vessel. These instrumentation thim bles are subjected to the cross-flow produced as the reactor coolant flows from the thermal shield area to the fuel assembly inlets. Fluid-elastic insta bility must be considered in essentially the same manner as for the hea t exchanger tubing. Determination of the flow field in the lower plenum of the reactor vessel via a fluid mechanics computation (e.g., porous media approach) is the first step in this analysis. 3.2.4
H ydraul i c Design of Control Rods
To obtain a simplified method of analysis for estimating the drop time of a control rod, the basic assumption is made tha t all the hydraulic forces are proportional to the square of the rod velocity. The equation-of-motion of the control rod is then M+m dV = I. F gc dt
(3.64)
--
where M is the mass of control rod and m is the virtual mass of the body due to fluid acceleration; i.e., m = M p l / PM
(3.65)
The equation-of-motion can be simplified to
(3.66)
dV - C2 - C3V 2 dt _
Solution of the above equation is 1 1 + [1 - exp( 2C3 x)fi2 � In � [1 - exp( 2C3 x)] 2(C2C3 ) where x represents distance traveled. Asymptotically, as x � (x) (or x � 2 /C3 )' t-
_
1-
-
-
1/2
C3x.:;.... +---: In2 :t = ----.(C2C3 )
(3.67) (3.68)
At the end of the rod's travel, it enters a dashpot with a small clearance between it and the rod. This clearance produces a high frictional force that rapidly decelerates the rod. The resisting force in the dashpot is linearly proportional to the distance of insertion, Z. The equation-of-motion when the rod enters the dashpot is
(3.69)
H Y D RO D Y NAM ICS
283
The solution to this equation is v 2 = vfJ exp( - 2C3 Z - C4 Z 2 ) + 2 exp( - 2C Z - C4 Z 2 ) 3 X
c2
f
z o
exp(2C Z + C4 Z 2 ) dz 3
where Vo is the entering velocity. This can be rearranged to V 2 = V � exp(C � /C4 ) exp( - u 2 ) u + (2C2 / � ) exp( - U 2 ) exp( u 2 ) du
fCJ/V'C4
where
(3.70) (3.71)
U = C /� + �Z 3 Hence, rod velocity in the dashpot can be evaluated at each Z. An additional concern in the hydraulic design of rod cluster control rods is the vibration of the rodlets when the control rod is in the fully withdrawn or nearly fully withdrawn position. Such vibration leads to wear and even tual penetration of the tube containing the poison material. The upper portion of the control assembly is contained in a solid guide tube and is thus protected from fluid buffeting. However, in the region just above the fuel assembly, a solid guide tube cannot be provided because provisions must be made for the exiting coolant to leave. In this region, where the coolant is flowing out of the assembly and across the core to an exit port, separate partial shrouds are provided for each rod let. The shrouds are usually hollow tubes with a portion of the back side cut away to allow access for the spider. The open area means that the rodlets feel some effect of the cross-flow in this region. Since the tubes are not fully exposed to the cross-flow stream, the equations for the previous section are not directly applicable. Because of the complexity of the geometry, it would appear that a determination of whether a proposed design is satisfactory requires ex perimental verification.88 3. 3
TWO-PHASE F LOW
Although there is no net void at the outlet of a PWR under normal oper ating conditions, there is substantial subcooled boiling within the cores of modern plants. Bulk boiling is allowed in the hottest channels; further, dur ing most accident situations, boiling occurs throughout most of the core. Neither steady-state nor transient behavior can be understood without a knowledge of two-phase flow fundamentals. Since the details of two-phase flow behavior are available in a number of references (see Refs. and 91) , only a summary is presented here.
89, 90,
TH ERMAL A N A L Y S I S OF PWR S
2 84
As noted previously, the discussion of this chapter is confined to steam water behavior and does not deal with two-phase flow situations that might be encountered in severe accidents. 3.3.1
Two-Phase Flow Structu re (Flow Patterns)
When liquid and vapor flow together in a channel, there must be an inter face between the phases. In most situations, the flow takes on a particular structure or configuration. The various configurations seen are generally designated as The typical flow patterns encountered in co current flow through a hor izontal tube are illustrated in Fig. 3.1 3(a). At the lowest gas and liquid flow rates, the liquid flows along the bottom of the tube and the gas along the top, which is designated stratified flow. At higher gas flow rates, waves are found on the gas-liquid interface; this is called At higher liquid flow rates than those that occur during stratified flow, we encounter if the gas flow rate is very low. Here, small bubbles of gas tend to flow in the upper portion of the tube. When the gas flow rate is increased, large bullet-shaped bubbles, called are formed. In plug flow, these plugs move through the liquid along the upper portion of the tube. An increase in gas flow leads to where frothy slugs of liquid move rapidly across the upper region of the tube. Between the slugs there is a wavy layer of liquid at the bottom of the tube. At high gas velocities appears. Here the liquid flows around the outside of the tube while the gas flows in a central core. Further in creases in gas velocities lead to entrainment of some of the liquid as drop lets carried along in the central gas core. At very high mass flow rates, we encounter dispersed flow. At low qual ities this is seen as a froth of tiny bubbles essentially uniformly distributed in the liquid. At high qualities it is seen as a mist of fine droplets suspended in the vapor. A significant change in flow patterns occurs when a line is inclined slightly to the horizontal. The stratified flow pattern disappears and is re placed by intermittent flow. Weisman and Kang9 2 suggest that stratified flow will disappear when
flow patterns.
wavy flow.
bubble flow
slug flow,
plugs,
annular flow
sinS > D / L
(3.72)
where S is the angle of inclination and L is the length of line being examined . Wavy flow is still seen in inclined lines, but the wavy flow region is restricted. As the inclination angle increases, the gas flow rate at which wavy flow begins also increases. At a sufficiently sharp angle of inclination, wavy flow disappears altogether. However, the bubble flow area expands as inclina tion is increased .
H Y D R O D Y N AMICS
285
Bubble
Slug
Plug
A n n u l o. r
Strati
l e ol
D i s pe r s e ol
low 'W a v y
(a)
o 0
o
.
•
•
• ·
•
Plug
Bubbly
Fig. 3.13
•
•
o
(b)
.
•• . . . , . . . . . . . . . •
o
•
. ' .
Churn
Annulnr
•
•
• • •
•
•
• •
• • '
• . •
• •
•
D i s p e r s e o/
Flow patterns i n (a) horizontal flow and (b) vertical flow [from T . Crawford, Ph.D. Thesis, Drexel University, Philadelphia, PA (1983)].
In vertical lines, and lines at very sharp angles of inclination, slug flow disappears and the only flow patterns seen are bubbly, plug, churn, and Churn flow is a chaotic mixture of large packets of gas annular (Fig. and liquid, which seem to have a churning motion. Since flow patterns can affect pressure drop and clearly will affect inter phase mass and momentum transfer, it is desirable to be able to pred ict the
3. 1 3).
T H E RMAL ANALYS I S O F PWR S
286
1 0 0 0 �-----'��
1 00
�
AN N U L A R
10
t;,j --
�
0
� Cf)
-e--
Il::
';;;,.'!f .
� Il. Cf)
-
0
0. 1 BUBBLE
0.0 1
L-__L-��-U�L-__L-J-���----�����
0.01
Fig.
3 . 1 4(a)
0.1
10
Generalized flow pattern map for horizontal flow.
flow pattern that will exist at a particular set of conditions. Early attempts at correlating the transitions from one flow pattern to another93 attempted to use the same dimensionless groupings for all transitions. Subsequent analyses94,95 have indicated that this is inappropriate and that separate mechanisms govern each transition. Despite this complexity, simplified flow maps in terms of superficial gas and liquid velocities have been de vised.92,95,96 The steady-state maps applicable to horizontal and vertical flow are shown in Figs. 3.14(a) and (b). The fact that different mechanisms govern the individual transitions is accounted for by plotting the maps in terms of (VSC /
287
H Y D R O D Y N AMICS
1 0 0 0 �------ ----- -� ��
1 00
AN N U LAR
G �
10
---
�
ff ---
';:;..�
0. 1 BU BBLE
0.0 1
L-__L-��-U�L-__L __��-U�L-__L-����
0.1
0.01
Fig.
3 . 1 4(b)
10
Generalized flow pattern map for vertical flow.
correction factors obtained from Table 3.111. To simplify the maps, the slug, plug and churn flow patterns have all been combined and considered to be "intermittent" flow. Crawford et a1. 96 found that the flow maps of Fig. 3.14 could be extended to inclined and downward flow by appropriately modifying 1 and 2 for these cases. These modifications are shown in Table 3.III. Note that since Fig. 3.14(b) is used for both upward and downward flow, the unstable region, seen at very low liquid velocities in downward flow, is not shown. This region occurs when the buoyancy-induced upward velocity of the va por is equal to or exceeds the downward velocity of the liquid. Crawford and Weisman97 have compared the flow pattern map of Fig. 3.14(b) for vertical flow in simple tubes under adiabatic conditions to data
b) Downward
a) Horizontal and upward
(r:;) 0. 1 9
0.8
[0.65 - 0.3
s
(g ) 8
co s )
0.6
[ Cl.7 (cOS8)0.3 ] ( �:�) 3 ] - 0.65
1 + 4(sin1 .95 i 8 i )
�
1 .0
0 Os
0 Os
1 .0
(�J" (-r·45 1 .25
Os
Q>2 .4 ( PLS) 0.33 (Q) 0.21 ( ��LLS) 0.09 (�CTS) 0 2
*S
denotes standard conditions. 0, = 1 .0 in. = 2.54 cm, Pcs = 0.0013 kg/I, P LS = 1 .0 kg/I; i-LLs = l cP; CTs = ·07 N/m (70 dynes/em) � P = P L - Pc Source: Based on T. J. Crawford, Ph.D. Thesis, Drexel University, Philadelphia, PA (1 983).
Bubble-intermittent
[
C :�8 i ) ]
separated flow disappears
b) Sharply inclined e) Downward exp
1 .0
OS
a) Horizontal and slightly inclined
Pc
Intermi tten t-separa ted
� Ps
Os
All
P LS
�
To annular flow
Pes Pc
O. 9 .24 ( ) ( ) 0.67 (Q) 0.21 ( �LS� ) 0 (�CTs ) 0 45 ( PGS) 0.23(�) O.1 1 (�CTS) O. 1 1 (Q) 0. 1
Q>1
All
Flow Orientation
To dispersed flow
Transition
TAB L E 3 . 1 1 1 Property and Pipe Diameter Corrections to Overall Flow Map*
� ;;0 U'l
""'0
""T1
-< U'l Vi 0
r-
» z »
r-
»
�
m ;;0
-i I
t-.J co co
H Y D R O D Y NAMICS
2 89
obtained with adiabatic flow (with heating). They concluded that although considerable scatter was observed, most of the data were in reasonable agreement with the map. However, they recommended that the intermit tent-bubble correction factor be modified at high values of (VSL /
69
Bubbly / churn flow for a < a tran
+
Annular flow for atran < a <
(PI Pg ).
It may be seen The parameter a tran is a function of the density ratio, / that TRAC does not consider mass velocity effects. Under normal PWR operating conditions, the mass velocity is generally above 2 X 106 lb /hr ft2 At this velocity, and the water properties found at PWR temperatures, dispersed flow is to be expected in the hot channels.
T H E RMAL A N A L Y S I S OF PWRS
290
L � 0.9 1
0.8 0.7
a
�
'
M is t - - - - - - - - - - - ---.j - -
-
_
_·!'r� ny 0} �n - - J��n u�a� _
- �
-
_
Tra n s i t i o n -
-
-
-
- - - - -
, l
0.6 0.5 0.4
I
I
I I
0.3
0. 1 °
- -
Tra n s i t i o n -
-
-
-
Bu bbly
-
-
, r-- - - - - - - I
I Tr a n s i t i o n l
Slug (I n te r m i t te n t )
0.2
M ist ( D i s p e r s e d Dro p l e t )
-,
-
-
-
Tra n s i t i on
I i-
I
D i s p e rs e d B u b b le
--,I
�------�- � ------- ----
3 X 1 03
Fig. 3.1 5 Flow pattern map used by RELAP-5 [from Ref. 1 01 ] .
At somewhat lower velocities, bubble and intermittent (churn) flow would be encountered. Because of the relatively low vapor volume fractions at normal or near-normal operating conditions, annular flow would not be seen. However, annular flow could be encountered during the reflood phase following a LOCA. In addition to annular upflow in the core, countercurrent annular flow may be encountered in the downcomer during a LOCA. 3 .3 .2
Void F raction
a,
Because of the relatively low density of steam compared to liquid water, the steam vapor phase is called void. Void fraction, is defined as the ratio of local vapor volume to the total local flow volume, that is,
a = -g-
A Ag + A[
-
(3.74)
where A[ is the cross-sectional area occupied by the liquid and Ag is the cross-sectional area occupied by the vapor. The average density, p, of a two phase mixture then becomes
(3.75)
291
H Y D R O DYNAMICS
Reg i on '
Reg i on
Reg i on
1/
1/,
Di stance 'n F l ow D i rect ion Fig. 3 . 1 6 Void fraction in boiling region (from Ref. 1 04).
where Pg and PI are the vapor and liquid densities, respectively. Slip ratio 5 is defined as the ratio of vapor velocity to liquid velocity 5 - Vg
VI
X _ I - a Ei
_ _
I -X
a
(3 7 ) .
Pg
6
where X = flow quality mass flow rate of vapor / mass flow rate of mixture. Void fraction in a heated channel can be described with the aid of Fig. 3. 1 6. Initially, the entire fluid is highly subcooled and no void is present. When the liquid adjacent to the wall becomes superheated, bubbles form. In region I, where the bulk of the liquid is still highly subcooled, the voids are a thin layer attached to the wall. In region II, where subcooling is low, void fraction increases and bubbles are detached from the wall and recon dense very slowly. In region III, the bulk liquid has reached saturation and bulk boiling occurs. Maurer103 correlated void fraction in region I in terms of 0, the effective thickness of the vapor film on the wall. Thus, =
a=
and
0
is the maximum of
(3.77)
40 -
De
0.00033 1.07h (Tsat - Tbu1k ) ft or
0=
q;; kI Pr,
2
'
(3.78)
T H E RMAL A N AL Y S I S O F PWRS
292
where q;; = heat transfer rate due to boiling = q'�o ta I h ( Tsat - Tb ) h = forced convection heat transfer coefficient -
kl = Prl =
thermal conductivity of the liquid Prandtl number of the liquid.
a
The value of computed from Eqs. (3.77) and (3.78) is always very low and therefore it is often ignored. Early work by Bowring l 04 showed that the detached region begins at a critical fluid subcooling that depends on flow velocity and heat flux. Cor relations for the detachment point criterion have been presented by Bow ringlO4 and several other investigators. lOS-l OS Of these, the criteria of Levy 107 and Saha and Zuber 108 appear to be the most successful. The Levy l 07 model is based on the idea that at the bubble tip, which is located at dimensionless distances, YB+ from the wall, the liquid tempera ture should be as warm or warmer than the saturation temperature. The departure criterion is then established by assuming that the liquid temper ature follows Martinelli's universal temperature profile and exceeds the sat uration temperature at the departure point. The value of YB+ is given by (3.79) and the departure criteria are: if 0 '" YB+ '" 5.0 ,
Hf - Hid
if 5.0 � YB+ � 30.0 ,
= cp �' (fj'S) -
G
Hf - Hld = CP
q" h
- -
1;;
Pr YB+
5.0q " l G ( t/8) iz
{Pr + In[ 1 + PreyB+ /5.0 - 1 .0)]} if YB+ ;?: 30.0 ,
q"
-c Hf - Hld P h
5.0q" G ( t/ 8) 11z
where G
=
h = Hf = Hid =
(3.80b)
- - -----� -'-
([Pr + ln(1 .0 + 5.0Pr) + 0.5 In(YB+ /30.0)]} Cp =
(3.80a)
specific heat of liquid mass flux, mass/area time single-phase forced convection heat transfer coefficient enthalpy of saturated liquid enthalpy of liquid at point of bubble departure
(3.80c)
293
H Y D RO D Y N AM ICS
Difz === q" = Pr
IT
i-L[
friction factor hydraulic diameter total heat flux, energy / L 2 t liquid phase Prandtl number surface tension liquid phase viscosity.
= =
A somewhat simpler departure criterion, which appears to work about as well, has been presented by Saha and Zuber. l OB They concluded that at low values of the Peclet number, bubble departure was thermally con trolled, while at high Pe values it was hydrodynamically controlled. Their analysis yields the following criteria: for
q"D ) Pe�70,OOO, Tsat-TL =O.0022 (T Pe>70,OOO, Tsat -TL cpq"C
for where
Tsat = TL = = k[ = Pe
= 153.8 -
(3.81a) (3.81b)
saturation temperature bulk liquid temperature Peclet number = Re Pr thermal conductivity of liquid.
At fluid enthalpies above that at the bubble departure point, a significant void fraction may exist even though the liquid enthalpy is below that of saturated liquid. Lahey and Moody109 present a mechanistic model for com puting the quality in this region. The model is based on estimating the vapor produced by the fraction of the applied heat flux going to boiling and subtracting the condensation of vapor by the subcooled liquid core. Their expression has the form
1 _[ Z ( Pll qb1 ) dz- z p q" dZ ] X 1 (Z)= _ CAH I 1 +£ I fg
where
Xl
Zd
Zd
h cond
A= (z) Zd = =
flow area actual quality at location location at which bubble departure occurs heated perimeter
qi;
heat flux due to boiling and bubble agitation = q "
Ph
(3.82)
z (z > Zd)
(Z;=Z::)
T H E RMAL ANALYSIS OF PWRS
294
HI = enthalpy of liquid at location z = H(z) - x 1 (z)Hfx H(z) = total fluid enthalpy at location z " q = total heat flux H A " condensing heat flux = ---.i.E. q cond 1Jf8 Ph = enthalpy and specific volume change on evaporation, , Hfx 1Jfg
h o'
hb =
respectively condensation related coefficient = 1 50
(h
°F)
= PI (Hf - H/ )/[pg Hfgl . Note that once HI Hf , the liquid enthalpy remains at Hf and all of the heat £1
=
flux goes toward boiling. In view of the cumbersome nature of the foregoing relationship, simpli fied approaches are often used in design. A very simple approach is the so called "profile-fit" method where the actual vapor mass fraction x l(z) is related to the "thermodynamic quality," x(z). The relationship used is
[- ]
x 1 (z) = X(Z) - Xld exp X(Z) X ld - 1 where
(3.83)
X (z) =H(z)/Hfg Xld = ( H1d - Hf )/Hfg Note that Xld will be negative. Empirical correlations have also been used. Thorn al. 1 10 correlated void fractions in subcooled boiling for the 750- to 1 000-psi range. Their correla tion, which is in terms of apparent density p, appears somewhat simpler. Tong1 1 1 extended their correlation to 2000 psi; the modified correlation is
et
)I X ' ( P/ - Pg ) + X ' ()I ' - 1 )
P - P/ - 1 '
_ _
where
PI ' Pg = liquid and vapor densities, respectively X'
=
(Hin + 4LD q" - f3Hf)/(Hv - f3Hf ) eG
{
PI /( P v5) = exp 4.21 6[( y - 8.353)2 /(8.353)2 - l f/2 } tj = In ( P /3206)
)I
'
P
f3
=
pressure in psia ratio of enthalpy at inception of detached region to the G saturation enthalpy = 1 - 0.15 q " /
.
(3 84 )
H Y D R O DYNAMICS
Hin Hv , Hf
C L
=
5 =
=
=
295
inlet enthalpy to channel, Btu /lb saturation enthalpies of vapor and liquid, respectively, Btu/ lb mass velocity, Ib / (h ft2 ) slip ratio heated length, ft.
The above equation was developed from uniform heat flux data; although it has been applied to channels with nonuniform heat flux and mixing by replacing / by the enthalpy rise to the point in question, experi mental verification is lacking. When bulk boiling begins, quality is determined from fluid enthalpy H by
(4Lq" De C)
X
(3.85) Void fraction is obtained from the quality. However, void fraction and slip ratio during bulk boiling vary with the flow pattern. At the pressures and flows encountered in a PWR, the flow pattern existing at the onset of boiling will be small vapor bubbles dispersed through a liquid core (dispersed or bubbly flow) . At higher void fractions, small bubbles may coalesce into large elongated bubbles, and churn flow may be seen. If enthalpy is in creased beyond the range normally observed in a PWR, the vapor forms a continuous core with an annulus of liquid adjacent to the wall (annular flow). The simplest treatment of dispersed or bubbly flow is Bankoff' s.u 2 By assuming a power law distribution for velocity and void fraction, the fol lowing equation relating x and a was derived:
1/ X - PI Pg =1
/
(1
K ' / a)
(3.86)
with the flow factor K defined by
K ' = a + (1 - a) / S
(3.87)
Bankofp 1 2 proposed that constant K ' can be obtained from
+
P
(3.88) where is in psia. Revised values for K ' were proposed by Kholodovski l 13 and Hughmark. 1 14 By correlating available data, Hughmark concluded that K' was a function of parameter Z, where K ' = 0.71
P
Z=
Ph
0.0001
)1;8 (1 X)P +XP/] [ /--LI -De C+ ] 1J6(� p]; gDc [ _(1 - Xg)Pg 'I4 (1 /--L/ ' /--Lg a)
/--Lg a
(3.89)
wh ere = mixture density obtained by assuming no slip and = liquid and vapor viscosities, respectively. All other symbols have their previous mea ning. The relationship between K ' and Z proposed by Hughmark is
T H E RMAL A N A L Y S I S O F PWRS
296
TAB L E 3.1V Relationship Between Flow Factor K and Parameter Z
Z
K
1.3 1 .5 2.0
0 . 1 85 0.225 0.325
Z 3.0 4.0 5.0
K
Z
K
Z
K
Z
K
0.49 0.605 0.675
6.0 8.0 10
0.72 0.767 0.78
15 20 40
0.808 0.83 0.88
70 1 30
0.93 0.98
given in Table 3.IV Use of Hughmark' s relationship leads to slip ratios that approach 1 .0 as mass velocity increases. This is in accord with experimental observations. The Hughmark correlation tends to give poor results for al phas (a) greater than -0.8. At high a , it is probably preferable to obtain the relationship between a and X by simply extrapolating the Hughmark X versus a curve developed for a < 0.75 so that the curve goes through a at X = 1 .0. More recently, Zuber and Findlay 1 1 5 proposed a more sophisticated model which can now be applied to all flow patterns, although it is most useful in the bubble, dispersed and intermittent regions. They define local superficial velocities u I A), and u/ A) based on I A, U V = total flow area A and volumetric flows, Velocity and void distributions were assumed to follow
=1
= (Qv + QJ)
Q.(Qv
�: =l- GJ
a-a
ac -
R
� =1-
(r) n R
= (Qd
(3.90a) (3.90b)
where r is radial distance, is tube radius, and subscripts c and w refer to centerline and wall conditions, respectively. For adiabatic flow, aw is taken as zero. If locally the vapor were to have the same velocity as the liquid, there still would be a difference between the average vapor and liquid velocities since the vapor is not uniformly distributed over the cross section. Under the condition of equal local vapor and liquid velocities, we may define a distribution factor Co such that:
I3=Qv/(Qv+QJ )' 11K'
=13 1 _, = n+2 n+ l Q Q/
Co
a
-
(3.91 )
where with V I as the volumetric flow rates of vapor and liquid, respectively, and a ' as the mean valuE' of a assuming locally equal vapor and liquid velocities. Note also that with equal local velocit ies , Co equals in the Bankoff1 1 2 and Hughmark1 14 void models [Eq. (3.86) ] .
H Y D R O DY N A M I CS
VI" Vg =CuU+Ugl Vg Uv / a ugi = j l ' drift flux: g jgl
297
In actuality, the local liquid and vapor velocities are not identical and the true average vapor velocity, is given by
(3.92)
where = weighted mean drift velocity of vapor. It is also and usual to define a quantity, called the
(3.93)
Drift flux represents the volumetric rate at which vapor flows through (forward in upflow, backward in downflow) a unit area of a plane normal to the channel axis and moving with the flow at a velocity This approach to void estimation is commonly referred to as the The value of the drift velocity, is closely related to the terminal rise velocity of a bubble through the liquid and it is often assumed to be iden tical with this rise velocity. The value of is positive in vertical upflow, negative in downflow, and often taken as zero in horizontal flow. Once the are specified, the average void fraction, may be com value of and puted from
UgI'
drift fluxu.model.
ugi
Co ugi
ugi Co Ugi [(CoG/ P/ )+ugd p/ (1 -a) (1 -aCo)[G-apg ugl /(1 -aCu )] Co(3.90) Co 1 Co> 1 Co Co = 1 Ugi = O. Co ugI ,
a,
(3.94)
At high pressure (typical PWR conditions), is small and the exact value used will probably not affect the results significantly.1 16 The slip ratio, S, may be determined if and are known by 5
=
(3.95)
Early attempts at defining generally were based on determining the appropriate values of n in Eq. to be used with the various flow pat terns. It can be shown that < when the void concentration at the wall is greater than in the central region and for the reverse conditions. For homogeneous flow, and Most recent applications have tended to regard simply as an empirical quantity to be fit to the available data. Chexal and Lellouche1 17 have developed a very comprehensive set of exp ressions for which fit a wide variety of data. However, the and model is complex and cumbersome to use. A much simpler model, which has been shown to perform well, 1 1 8 is that of Dix 1 19 as modified by Lahey and Moody. 1 09 In this model
(3.96)
whe re b
= (pg / p
I
)0. 1
T H E RMAL ANAL Y S I S O F PWR S
298
Ugl = 2 . (J e
9 [ (P/- PPgI)(Jggc ]
(3.97)
. 114 SIn e 2 = surface tension = angle of inclination (positive if upward).
et
Chexal al.11 9 indicate that the model generally performed well except in large pipes. Kawanishi al. 1 20 present a more complex set of relationships for and UgI ' which are, however, less complex than the Chexal-Lellouche equa tions. 1 17 The proposed relationships are provided in Table Kawanishi al.1 2o compare their correlation to a wide variety of upflow and counter flow data and obtain better agreement with the data than with the Chexal Lellouche approach. The drift-flux model is now the most widely used approach for high-velocity forced convection systems. However, it appears to provide
et
Co
3.V
et
(1994)
TA BLE 3.V D rift Flux Parameters of Kawanishi
Average Velocity of Mixture u (m/s)
Pressure P (MPa)
Tube Diameter o (m)
0 < u < 0.24
P � 18
D � 0.61
u ;:o: O.24
P � 1 .5
D � 0.05 0.05 � D
1 .5 < P � 1 8
o � 0.02 0.02 < 0 < 0.24
et al.1 2 0
Drift Velocity
U81
Interpolate
{g
0
(PI - Pg )
0.52 ----"'PI Same as P � 1 .5 MPa Interpolate
}
0.24 � 0 � 0.46
Average Velocity of Mixture It (m/s) u � - 3.5 - 3.5 � u � - 2.5 - 2.5 � u < 0 O�u
Distribution Parameter Co
1 .2 - 0.2\!Pg IpI 0.9 + 0.1 \!ps I PI - 0.3(1 - \!Pg I PI )(2.5 + u) 0.9 + 0. 1 \!p,,/PI 1 .2 - 0.2\!ps IpI
1/2
299
H Y D R O D Y NAMICS
less satisfactory results at low velocities and natural circulation conditions. At the low velocities present during the reflood phase of a LOCA, Yeh and Hochreiter1 21 recommend
0.925(pg / P ) ( uer) ( u" + HI ) 1.53g1/2(gc
=
/
O.
239
Uv
b �
O.
(3.98)
6
where
Uer
UV '
=
if U v Uer < if superficial velocities of vapor and liquid, respectively.
For natural circulation systems operating between and bar, Moch izuki and Ishi 1 22 recommend that the void fraction be obtained from the slip ratio, where _
_
1
(3.99)
1
and = tanh(5x) + In the annular flow region, the effect of mass flux appears to be relatively small. The void fraction relationship of Martinelli and Nelson, 1 23 which is presented as a plot of ex versus quality, for various pressures, has therefore often been used. Thom 1 24 modified these curves on the basis of improved data and proposed a correlation of the form
= 1 +X()'"1-"--1 x1 -1)
ex
--
(3. 1 00) /(P/5).
--
x
3.VI.
where is flow quality and )' is an empirical constant equal to P 1 Thorn proposed the values of )' shown in Table 1 All of the relationships between and ex indicated in this section apply strictly to steady-state conditions only. During rapid transients, the rela tionship between x and ex may not be identical to that seen in the steady state. This question is considered in Sec.
x
3.3.4.3.
TAB L E 3.VI Parameter Values for the Thorn Void Fraction Corre lation
Pressure (psia) 'Y
1
14.7 246
1
250 40.0
1
1
Values of 'Y
600 20.0
1 250 9.80
1
2100 4.95
1
3000 2.15
1
3 206 1 .6
T H E RMAL A NA L Y S I S OF PWRS
3 00
Two-Phase Pressu re D rop
3 .3 . 3 3.3.3.1
Pressure Orop in Pipes and Rod Bundles
In the highly subcooled boiling region where only attached wall voidage is present, it has sometimes been assumed that single-phase pressure drop calculations are adequate. However, this is not the case at high heat fluxes typical of reactor cores. The measured pressure drop is significantly above the single-phase value. Correlations of subcooled boiling pressure drop were proposed by Owens and Schrock,125 Le Tourneau and Troy, 1 26 and Tarasova et al. 1 27 The correlation of Tarasova et al.,1 27 which includes both friction and accelera tion losses, is particularly convenient. Tarasova gives the ratio of boiling pressure drop, (dP / dZ)/J ' to isothermal (nonboiling) pressure drop, (dP / dz)o, as (dP / dZ)/J / (dP / dz)o = 1 [20N / (1 .315 - N)] x [ q Vg / (Hfg Vin )] 0 0 7(Vg / VI ) 0 0 08 , (3.101)
+
0,
where
"
(H - HO ) / (Hsa t - HO )
N
" q = wall heat flux, Btu/h ftl
specific volume of gas and liquid, respectively, ft3 /lb liquid enthalpy, Btu/lb Hsat t HO = liquid enthalpy at saturation and inception of boiling, respectively, Btu/lb = inlet velocity, ft/ h Vin enthalpy change o n evaporation, Btu /lb. Hfg Vg , VI
H = =
The foregoing correlation was developed for data with heat fluxes between 0.18 to 0.55 x 1 06 Btu/h ft2, pressures between 142 and 2840 psi, and G between 1 .0 and 2.2 X 1 06 Ib /h ft2. In the detached and bulk boiling regions, the increase in fluid volume significantly increases the velocity of the stream and, therefore, its momen tum; consequently, all three pressure drop components indicated by Eq. (3.5)-frictional loss, momentum change, and elevation pressure drop need to be considered separately under these conditions. At the low void fractions < 0.30) and high mass flows usually encountered in a water cooled reactor, two-phase pressure drop can be evaluated from the homo geneous model proposed by Owens.128 The elevation pressure drop is given by
(a
6. Pe1ev =
fop L
d L=p6. L
(3.102)
For a local boiling flow, p can be obtained from Eg. (3.94). The friction pressure drop is obtained from
301
H Y D R O DY N A M I CS
d Pfriction =
fd L v c 2
De g 2
(3.1 03)
c
and the acceleration pressure drop term is obtained from C 2 dv
gc
(3. 104)
d Pacc = - - d L dL
where C is the mass velocity (lb/h ft2) and v is the average specific volume of the mixture at the location considered. Evaluation of friction factor f in Eq. (3.1 03) requires estimating a Reyn olds number. Owens1 28 simply used the liquid viscosity here but other in vestigators have proposed using a two-phase mixture viscosity. In some cases, the formulation suggested by McAdams et has been used:
al. 1 29
(3. 105 ) where f.1s ' f.1, = viscosities of gas and liquid, respectively, and "jl: = viscosity of two-phase mixture. However, Choe1 30 obtained improved agreement with available data by assuming that the viscosity of a bubbly mixture is actually higher than that of a pure liquid. Choe 130 suggests that fluid vis cosity can be obtained from the following: /1: = f.1 , exp[2.5 / (1 - 39a/64)]
a ::::; 0.5
or a > O.5
(3. 106)
where f.1c = /1: at a = O.5 The pressure drop obtained during high void-fraction flow again de pends on the flow pattern; however, a general empirical correlation for calculating the two-phase frictional pressure drop was developed by Bar oczy. 131 His work is considered an extension of that of Martinelli and Nel son 1 23 and Lockhart and Martinelli 132 who defined the two-phase pressure drop in terms of a single-phase pressure drop. Baroczy writes (3.107) where
302
T H E RMAL A N A L Y S I S OF PWRS
6000 4000 2000
0
"S-
Ui
� 0 Q
-
�
�
0
...c a... ,
0 � I-
1 000 800 600 400 200 1 00 80 60 40 20 10 8 6 4 2
O. J
1 0.00001
0. 1 0.001 Property Index, (iJ/ll ) o.2 / ( p/ lpv ) Sodi um (O � koo L ' ---....L'OO Freon -22 ( OF ��L..... 0.1. 0-2--'0'00 .1. 8.a... !0-"1!l �'-0-1....800--.J .. 0-2....0.. 5 6. �1 4 --80:-::' 0 -1.... 1 6-'Potassium F) 0 (° 1 000 1 200 1 400 1 600 1 800 20 0 Rubidium (Of) 1 000 1 200 1 400 1 600 1 800 Mercury (oF) 800 600 1 000 1 200 1 400 ' ----:-" Water (OF) L 705 21 2 328 467 545 636 .0.0001
'__'-' _...L. ' --.J ' L ' __....J.
___----', ___..L. '_ .1. _'-' ___ ' _....
Fig. 3 . 1 7
Two-phase friction pressure d rop correlation for G = 1
131 1.
X
l Ob lb / (h ft2) [ from Ref.
By defining a property index [ (f-Ld f-Lv ) 0 .2 (p d Pv ) ]' Baroczy1 3 1 obtained a cor relation for
(f-LI / f-Lv)O. 2 (3. 108) ( d P / d L)g (PI / Pv ) and (d P / d L)g = single phase pressure drop obtained at the same mass ve locity, when fluid is entirely vapor. His correlation is given in two sets of curves:
G,
1 r2
(d P / d L)1
1 . a plot of the two-phase multiplier
303
H Y D RODYNAMICS
2. plots of a two-phase multiplier ratio as a function of property index, quality, and mass velocity as shown in Fig. 3.18. The latter ratio multiplies
=
1
(Pg PI
(3.109)
x)
where X = Martinelli parameter 1 )1/2 [(1 - 1 X ] O.9 ( �1 /f.Lg )O.l for fully turbulent liquid and gas flow. The parameter e depends on the mass ve locity and the Baroczy property index as shown in Fig. 3.19. Note that Fig. 3.19 is given in terms of r, while Fig. 3.17 is in terms of 1 / r 2 An alternative approach, which shows good accuracy when ( � 1 �g ) > 1000 and G > 100 kg/ m 2 s, is provided by Chisholm' s correlation =
I 34:I
(3. 1 10)
where B is determined from Table 3.VII(a) and 11 = 0.2 for axial flow. The values of
et
(3.1 1 1 ) where q " is given i n Btu / h ft2 and G is expressed in lb /h ft2 (for 710 < P < 2840 psia, 0.37 X 1 06 < G < 1 .9 X 1 06, and 0.032 X 106 < q " < 0. 53 X 1 06). The frictional pressure drop during boiling should be calculated in a stepwise manner for length steps for which the changes in quality and fluid properties are only moderate. At high void fractions, use of the homogeneous model for computation of the acceleration pressure drop is not desirable since the effect of slip can be sig nificant. It is convenient to write the accelera tion pressure drop as d Pa cc = r
G2
gcPI
(3.1 12)
Where r is a dimensionless acceleration multiplier. When the inlet fluid contains no voids, r is given by
304
T H E RMAL A NA L Y S I S O F PWRS
16 I� 14 13 12 I I
09 08 07 0
� C a:: a:: w :i Cl. � -' ::::l �
lX
C X Cl.
i
I-
. 2� " J ""
��
0 -
. 2 .. .., .
� ., :�
.J'�
06
O� 04 18
17 16 I � 14 1 :\
Q
"-
It> N eT
O !' .. £ 0' O !!
I Z I I
09 08 01 06
4 0 10 60
o. � �______________�________________�______________��________________ 0 001 0 01 0 0001 0I
Fig.
3.1 8 Mass velocity correction versus property index [from Ref. 1 3 1 ] .
305
H Y D RO D Y NAMICS
50
u
.� .� -
--
�
2 G=34-0 kg / m s - A 2 G = 6 8 0 kg / m s - B G = 1 3 6 0 kg / m2 s - C 2 = 2 7 10 Kg 1m S - D 2 G = 4 0 7 0 kg / m s - E ----
______
B C
A
D
10
Q) o
U
1 1�
i I
. .�
---�_��_�� I � I� I �__
10
100
P h ys ical Prop e r t y Paramete r ,
r
C of Eq. (3. 1 09) as a function of mass velocity, C, and property index [from Ref. 1 33].
Fig. 3.1 9 Coefficient
TA B L E 3.VI I Constants for Ch isholm's Two-Phase Multiplier Equation
(a) Axial Flow
f � 9.5
9.5 < f < 28
C
(kg/m2s)
4.8 2400/ C 55/C o 5 520/ (fCIl 5 21 /r 1 5000/ (f 2COS )
B
11
? 28
(b) Cross-Flow
'--
Flow Pattern Vertical, bubble, dispersed annular Horizontal (side to side), bubble, dispersed annular Horizontal (side to side), separated
B
< 500 500 < C < 1900 ? 1900 � 600 ? 600
1
0.37
0.75
0.46
0.25
0.46
T H E RMAL A N ALYSIS OF PWR S
3 06
r = ((11 -X- ae)e)2 + Xa;c ( PPg/ ) - 1 (3.1 1 3) where X c = flow quality at channel exit and ac = void fraction at channel exit. By proper evaluation of a, the effect of slip is included. The value of r is independent of the manner of heat addition (sinusoidal, uniform, etc.) sat
along the channel. 3.3.3.2
Pressure Orop Across Area Changes, Restrictions, and Fittings
Pressure differences across area changes can be computed by one dimensional momentum balances. For an abrupt expansion, Lottes1 36 re ports that the total pressure difference can be obtained from
Me = ��,2 ([:� x 2 (a�a �J ] -
(3. 1 14) where
au a2 = expansion void fractions upstream and downstream, respectively, of a = area ratio (small area /large area) G = total mass flow rate, mass/time area liquid and gas densities, respectively PI' Ps == mixture quality. X Janssen 137 gives the pressure loss across a contraction as = !l 2gcPI (�a 2{ £Lx C2a1 � _�) Ps 2 a 1 ( _ a� Ll P
c
-2
PI a2 { Pg
-x
2 (Ca31 a12 ) +(1 - X )2 [ C(1 -1 ( ) 3 - - -
307
H Y D RODYNAMICS
where
C= a I ' a2 )/2 al == (al(a3+a 2 a23 = + (2 )/2 a
vena contracta area ratio void fractions upstream and downstream, respectively, of contraction
void fraction a t vena contracta.
al a2
On the basis of an extensive analysis of available data, Weisman et al. I JR concluded that for abrupt expansions, and should be evaluated by assuming slip flow. They recommend Hughmark' s 1 1 4 relationship for ob taining from For abrupt contractions, they again recommended that be obtained by assuming slip flow and Hughmark's correlation.1 14 and However, substantial mixing occurs at the vena contracta and
X.
a2 a
aslip = a,
al
(3. 1 16)
which would be computed for vena contracta size pipe in where the absence of contraction:
K'K' = 11.5 -a 2
aa22 >::S 0.0.55 (3. 1 15) 2 2 ][ X O-X) ] 2 1 [( C ) aPe=20" 2g-e --1 C +O -a) -Pg - -PI =
if if
It was also found that assuming homogeneous flow everywhere provided nearly as good a correlation of the data as Eq. did. Then, total pres sure drop across a contraction can be approximated by
(3.1 17)
Although this equation does not represent what is actually occurring, it is a useful tool. When contraction and expansion are well separated, the total pressure loss is
(3. 1 18) aPe apt' (3. 1 16). (3. 1 14) (3. 1 15) a) (LID) 2, a3 (3.116) K' (3.1 19) K' = C' (LID)b K' C' 1.148 - 1.57a, b K'= l.5a -0.34. 1
where and are obtained from Eqs. and and is obtained from Eq. However, for restrictions with a length-to diameter ratio of < only partial mixing at the vena contracta is obtained. For these conditions, can be estimated from Eq. with mixing factor calculated from where = of above
and or below zero,
If the foregoing leads to a value is set equal to or zero.
T H E R � A L A N A L Y S I S OF PWRS
3 08
When the vena contracta is outside (beyond) the restriction (see Table 3.1 for vena contracta location), then a revised momentum balance must be used and we obtain
�:;� d { :; (�
t. P, =
2
- 2 IT
X 2 ",
�-
P , ( -- rrc ) C [ -x Pg 2 1 U 3
-
u2
+ (1
-
{ 0 _1"3 )2 (1
+ 0 - d O - "3
X )2
( 1 -1 u3
-- - --
1 - u2
where U , U2 = void fractions upstream and downstream, respectively, of 1 restriction u = void fraction at vena contracta 3 = vena contracta area ratio IT = ratio of restricted flow area to full flow area.
C
For short restrictions (L/ D � 0.5), there appears to be little mixing at the vena contracta and u can be estimated, assuming slip flow; e.g., from Hugh mark' s correlation.114 A t L/ D ratios above 0.5, some mixing at the vena contracta is obtained and K the mixing factor at the vena contracta, can be estimated from Eq. (3.1 19) with ' = - 1 .8u + 1 .05 and = 3.1 (u - 0.36) . Again, if this leads to a value for K above 1 or below zero, K is set at 1 or zero. Chisholm has correlated the available data on two-phase pressure drop across bends.1 39 His later correlation 139 is based on the use of Eq. (3.1 10) with a redefinition of r 2 and B. If we assume that the single phase loss coefficient K[ [see Eq. (3.9)] is the same for both liquid at gas flow, then for 90-deg bends (3.121 a) I
C I
b I
and B 90 = 1 +
2.2 K[ (2 + R / D)
(3.121b)
where R = bend radius and D = pipe outer diameter. For bends other than 90 deg, Chisholm recommends Kl90 Be = 1 + (B90 - 1 ) K1 e
(3.122 )
where K'90 ' K1e are the single-phase loss coefficients for bends with angl es of 90 deg and S, respectively. Based on limited data, Chisholm and Sutherland 139 suggested that the two-phase pressure drop across tees be modeled using
3 09
H Y D R O D Y NAMICS
(3.123) where
C
]
C
]
+
[1 - ( 2 - 1 ) (vfg / vg ) 1i2 [ (vg / v I ) 1/2 ( vI / IIg ) 1/2 and C2 = 1 .75. More recent data have shown this to be a very considerable simplification of two-phase behavior in tee junctions. These experimental data show that appreciable phase separation occurs at tees1 40 and that mod els that treat each phase separately are required for good accuracy.141 Such computations are discussed in Sec. 3.5.6. Correlations of two-phase pressure drops across valves have been pro 1 43 for the intermittent and separated posed by Beattie142 and Simpson flow regions. Fairhurst144 proposed a model in which different multipliers were used depending on whether or not the flow quality was above a crit ical value. The tests of Kiederle and Mayinger 1 4S indicate that both the Beattie and Simpson correlations provide reasonable results for the separated and intermittent flows. The Simpson correlation, given by =
et al
.
et al.
et al. 1 43 �J)= { +x [ (:�r ] } { +x [ ( :J6 ] } 1
-1
1
-1
(3.124)
shows no liquid viscosity effect while that of Beattie 142 does. Since Kiederle and Mayinger's data also show no viscosity effect, Eq. (3.124) is recom mended. For the dispersed flow (homogeneous flow) region, use of the single-phase loss coefficient together with the homogeneous mixture density is recommended. Very little data are available in the open literature on the effect of two phase flow on the pressure drop characteristics of core spacer grids or wires. Since at reactor design conditions, the flow is generally in the dis persed region, a common approach is to use single-phase equations to gether with the two-phase homogeneous mixture density and velocity. 3 .3.4
Basic Two-Phase F low Modeling
The proper description of the steady-state thermal-hydraulic behavior of a nuclear reactor core requires an analysis based on a valid mathematical model of the two-phase fluid encountered in the hotter channels of the core. In addition, the appropriate description of reactor system behavior during accident situations requires a mathematical model for transient two-phase flow. In this section, the major approaches to two-phase flow modeling will be described and the basic equations used will be indicated . The discussion here will be confined to one-dimensional flow since most systems analyses aSSume that the flow can be taken as one dimensional. In Chapter 4, the
310
T H E RMAL A NALYSIS O F PWRS
revisions needed for the cases where multidimensional flow must be con sidered will be indicated. 3 .3.4.1
Single-Fluid Models
The simplest approach to two-phase modeling is to assume that a two phase mixture can be treated as a single fluid. A further simplification is obtained if we assume that a homogeneous mixture exists with vapor and liquid traveling at the same velocity. An analysis based on this approach would use the time-averaged governing equations, which are also averaged over the flow cross-sectional area. In their simplest form, the Eulerian ver sions of the differential equations for conservation of mass, momentum, and energy are identical to the single-phase conservation equations [Eqs. (3. 1 ) to (3.3)] except that and now represent mixture average veloc ity, enthalphy, and density and the frictional head loss, is based on the two-phase frictional loss. To obtain the values and at a given time, appropriate constitutive equations are required. When the homogeneous model is used, it is usual to assume that thermodynamic equilibrium exists and then properties are based on the appropriate thermodynamic tabulations. The model based on these assumptions is called the homogemeous-equilibrium (HEM) model. Computer programs that need thermodynamic properties find the use of tables time consuming. Simple equations that can represent these prop erties are greatly preferred. Garland .146 provide a highly accurate set of simple equations for this purpose. Lahey 147 has written the homogeneous flow equations in slightly differ ent form. Lahey' s model accounts for the effect of gravitation and can be used in a duct with a varying cross-sectional area.
V, H, p V, H, p
F,
et al
Conservation of mass: -( aza pV + aat Conservation of momentum: g!c [ acat (� a(C2az P) ) ] ----p azap gcg p V, A) A .-£ = 0
+
A/
A
(3.1 25) =
. Tw sln8 - Pf A
(3.126)
where the identity, G = has been used and A = cross-sectional flow area, area; Pf = frictional wetted perimeter, ft, (length); and Tw = wall shear stress, force/ area.
(pI) ap Conservatz.on of energy: p (aEat V aE) q" q '" + az J at , -+
-
=
- + A
1
-
(3. 1 27)
H Y D RO DYNAMICS
V 2 -g --) E= (H+-+ 2gc! gc J
where
z SinS
and q" q '" p'
J
= =
heat flux, energy / area time volumetric heat generation rate, energy/vol time the heated perimeter, length mechanical equivalent of thermal energy in 51 units).
(1.0
Single-fluid models can also be obtained without making the assumption of equal phase velocities. Tentner and Weisman1 48 derived a simplified, thermal equilibrium, single-fluid model in terms of the slip ratio. When we use the slip ratio, S, to define the phasic velocities, we can write
(3.1 28)
With this notation, and taking the cross-sectional area as constant, the equa tions for a horizontal system become: Mass conservation:
a-t[apg + (1 -a) P/ ] + [apg SVI + (1 -a) PIVI ] = 0 (3. 1 29) - [ap"SVI + (1 -a) PIVd + -[apgS 2 V? (3.1 30) +(1 -a)pVI I2 ]+ -= -a
Momentum conservation:
a az
a at
a az
"
ap az
T� A
+(1 -a)pIV[ EI ] --=ap q� at A
(3. 1 31)
The notation used in the above equations is:
Eg = Hg +S 22V 2 /(2gc!) E[ = H[+ V /(2gc!) VI' Vg = ,= q�
=
rate of heat transfer per unit length from wall to fluid, energy /length time liquid and gas velocities, respectively shear force per unit length exerted by the wall, force/length.
The foregoing model assumes that (as / az) and (as /at) are negligible and tha t enthalpy and internal energy differ by a negligible amount. The slip
T H E RMAL A N A L Y S I S OF PWRS
model would normally be coupled with the assumption of thermodynamic equilibrium. Such a model is designated as an unequal velocity, equal tem perature (UVET) model. 3 .3 .4.2
Two-Fluid Models
Two-fluid models are attempts to provide a reasonably complete and ac curate description of the flow of a mixture of different phases of one or more components without going to the usually impractical extreme of tracking the interface separating the volumes occupied by the different phases. Such models can allow for the significant differences between the velocities and temperatures of the gas and liquid (departure from homo geneous flow and departure from thermodynamic equilibrium) that may be present during transient boiling flows. Models of this nature are based on the use of separate conservation equa tions for the vapor and liquid phases. Thus the three conservation equation model resulting from the single-fluid approximation is replaced by a six equation model. The simplest of the available two-fluid models is that of Wendroff. 1 49 By defining the term a 1 and as
a2
the conservation equations for each phase can then be written in identical form. If SI units are used, we have
i= a(a·, p,V·, ) + a(a". p V,2 ) +a· -ap =Mz · at az ' az , a(a·, p, E·, ) + a(a·" p · E"V· ) +p (-'+--' aa . a(a v-'. ») =E· i= at az at az 1, 2
(3.1 32)
i = 1, 2
1
where
Ei r/ r;= -rn M zi Ej Ui Pi = Pie Pi ' Hi)'
[Uj + (V; g)]
= total internal energy density /2 ' = mass transfer rate to the i th phase, mass / time vol
(3.133) 1, 2
(3.1 34)
(note:
= rate of axial momentum transfer to the i ' tp phase, momentum / time vol = rate of energy transfer to i ' th phase, energy/vol time = internal energy, energy / mass.
If these equations are closed by specifying equations of state, say, and interfacial transfer rates, then there are six partial differ-
H Y D R O D Y NAMICS
ential equations and six state variables: V, a, VI ' V2 , HI ' H2 . These six equations provide one of the simplest two-fluid models and can be consid ered to apply to a variety of flow situations provided one can safely neglect viscosity, surface tension, and virtual mass. Nonequilibrium effects are in and E i · troduced by appropriately defining f i ' Wendroff' s basic model is perhaps the simplest computationally feasible full nonequilibrium model. The model is rudimentary in that the means of interphase transfer are unspecified. Further, the basic model is "ill posed" (differential equations are not hyperbolic) . Introduction of additional effects are required to obtain a well-posed system. Despite the fact that this basic model is ill posed, it is possible to obtain numerically stable computational schemes for it when a coarse mesh is used . The meaning of ill-posedness will be explored further in Sec. 3.3.4.5. More complex two-fluid models, in which interphase transfer mecha nisms are partially specified within the conservation equations, are availa ble. I so Unfortunately, most of these are also ill posed. Despite the fact that such an equation set may be ill posed, it has been possible to use such sets as a basis for numerical computation of two-phase system behavior. Several major transient analysis computer programs (e.g., RELAP5/Mod2I S I ) are based on an ill-posed model. Various schemes have been used to obtain a hyperbolic set of differential equations. The model used in the CATHARE code1 S 2 illustrates one such approach. A somewhat simpler approach is indicated by Lahey. I s3 He in corporates Reynolds stresses, surface tension, and virtual mass effects to achieve a well-posed set. The mass conservation equations are identical to those indicated by Eq. (3.1 32). He writes the momentum conservation equa tions as
Mzi
aa i a a aPi 2 - (a I· p I-VI· ) + -(a I· p I-V·I ) - a I· - + ( p I. - pI. )- + [I. V·I az az az at I
I
(3.135)
i = 1, 2
and the energy conservation equation as
a - V.I -(a ·T · ) + ( P dZ I R I
. -p . )
I
I
1
Di a i Dt
-
where D( ) / Dt = substantial derivation of ( ) with respect to phase ai
=
(3. 136)
i = 1, 2
interfacial area per unit volume, area/vol.
i
T H E RMAL A N A L Y S I S OF PWRS
314
= P; = Pi = TRi = q 'i
interfacial heat transfer rate, energy/area time interfacial pressure of phase force /area pressure of phase force/area Reynolds stresses in phase
i,
i,
i. Mzi ,
The rate of axial momentum transfer, is the sum of the effects of wall drag, interfacial drag, and virtual mass. For the liquid phase we would have , 1 Dt - Dt
Mzi=(Twp /A)+ g P/ Co I Vg -Vt l (Vg - Vt )ai -aP1 Cvl1 ( D.-.;Vg DIVI)
(3.1 37)
where
Tw = p' = A= Co = CVI11 =
wall shear stress, force/area perimeter flow area drag coefficient virtual mass coefficient.
A similar equation is written for the vapor phase but negative signs precede the virtual mass and drag terms. In the foregoing formulation, the phase pressures are not assumed to be equal but are related by
Rb =
Pg - PI = R1J - (pr !4)(Vg -V/)2 (2rr /
(3.1 38a)
where average bubble radius. The interfacial liquid pressure, defined by
P,�
pIJ '
is
(3.1 38b)
Pg.
but is taken as equal to The Reynolds stresses in the gas phase are considered negligible but the bubble-induced Reynolds stresses in the liquid phase are computed from
TRJ = S <Xt P/ (Vg -VJ )2 -1
(3. 1 39)
The equations used for phase pressure difference and Reynolds stresses in the Lahey model strictly apply only to bubble or dispersed flows. These should be replaced if the model is applied to separated or annular flow. The exact form of these terms and the virtual mass term is still a subject of debate (1 995). With an equation of state giving the density of a given phase as a func tion of the enthalpy of that phase and the pressure, the six conservation equations can be solved for and providing interfacial
Hg , HI ' Vg , VI' <X, P
315
H Y D RODYNAMICS
transfer rates are determined. Constitutive equations giving interfacial fric tion and heat transfer are therefore required for closure. A two-fluid model will provide greater accuracy than a single-fluid model only if reasonably accurate interfacial transfer rates can be established. Note that a two-fluid model does not assume, as required in a single fluid model, that the transient slip ratio be the same as that seen in steady state. If interfacial friction is properly modeled, the model will determine the appropriate slip ratio during the transient. 3 . 3 .4 . 3
Four Equation Models
The ill-posed ness problem and relatively long computing times associated with two-fluid models have led some investigators to the use of a simpler model that uses only four conservation equations. In the most widely used approach, differential equations for mixture mass, energy, and momentum conservation are coupled with an equation describing the time-dependent (dynamic) behavior of the phase velocity difference.98 In the so-called "dynamic slip model" used in the RETRAN program,98 the equation describing the time-dependent velocity difference (dynamic slip equation) is obtained by subtracting appropriate formulations of the liquid and vapor equations to obtain a single equation. This equation is then solved for the derivative of the slip velocity. We then have
a(VgI ) [ l (Ma )pc ] _ (�_�) ap pAgIBgIVgI Vg aVg at a2al P/Pg PI Pg dZ a1a2 P/Pg dZ +
=
+
+
(3. 1 40)
where
VgI = Vg -VI V = ul/(l -a) VgI = uv/a Vg = (aVg )/(a) l VI = «(1 -a)VI) / ( -a) = fA4>dA/ A = Vg =Vg , V/ =Vl) Ag/ A7I'g ' AWl Bg/ = 1 /8pcVglCg/ B Wi = 1 /8PiVi fi (4))
Uv
superficial velocity of liquid and vapor, respectively (note: In RETRAN, = liquid-gas interfacial area per unit volume = wall-gas and wall-liquid contact areas per unit volume interfacial friction coefficient = friction coefficient between wall and phase a =
T H ERMAL ANALYSIS O F PWRS
316
Ml1 = added (virtual) mass coefficient = b(1 - a)ap b = (O.5 + a)(1 - a) for a < O.5, b = (1 .5/a) - 1 for a ? O.5 fi = Moody friction factor between phase i and wall g I = interfacial drag coefficient r ' = rate of vapor generation per unit volume, mass/vol time Pc = density of continuous phase, mass /vol.
C
Equation (3.140) coupled with mixture mass, momentum, and energy con servation, forms a well-posed set. The dynamic slip model is usually coupled with the assumption of ther modynamic equilibrium. However, this is not a requirement of four equation models. Lack of thermodynamic equilibrium may be modeled by using the conservation equations for mixture mass, vapor mass, mixture energy, and mixture momentum. Alternatively, a liquid mass balance may be used in place of the mixture mass balance. By providing a separate vapor mass balance, one can incorporate nonequilibrium vaporization or conden sation rates. Models of this nature have been described by Ishi1 54 and Kroe ger 1 55 among others. The two-mass conservation equations coupled with the energy and mo mentum conservation equations allow the determination of x, FIg, HI , and the velocity of one of the phases, Vg or VI . The velocity of the second phase is determined via a constitutive equation specifying the velocity difference between phases. This specification has usually been accomplished by means of the drift flux approach (see Sec. 3.3.2) . Because of this, the entire model is generally referred to as a drift flux model. Drift flux models have the disadvantage that steady-state velocity dif ferences are assumed to hold during rapid transients. They have the desir able features of well posedness and fairly rapid computation times. Nevertheless, there is relatively little current (1994) interest in such models and most models of unequal temperature conditions use a two-fluid model. 3 .3.4.4
Constitutive Equations for Nonequilibrium Models
Closure for either the two-fluid or four-equation models requires the use of constitutive equations to define interfacial transfer rates. The form of these equations depends both on the formulation of the conservation equa tion in which it is used and on the physical model assumed for the inter facial transport process. Most modelers are agreed that interfacial momentum transfer may be obtained by exploiting the relationship between void fraction and interfacial shear that exists in the steady state. Crawford et al.100 show that if the void fraction a and the two-phase pressure drop, / are known for a given quality, then the product of the interfacial drag and interfacial area per unit volume in the steady state, may be obtained from solution of the
AgICgI,
� p �z,
H Y D RODYNAMICS
(3. 1 40)] -X)22 ] pg2 �p } 8 [a(1 -a)PgP,P 2 A Cg' {-=L [ foPg�p (1(1 -a) G Pg PI [(1 -x)apg -x(1 - a) P/] ppi (3. 1 41)
dynamic-slip equation [Eq. dependent terms set to zero: =
Sl
2Dpi
with vapor generation and time
+
+
where
AgI CgI p
f
=
= =
product of interfacial drag coefficient and interfacial area per unit volume = average density of mixture, Moody friction factor.
�P PI- Pg
(3. 1 41), AgICgI
It is assumed that the steady-state value applies to transient situations and hence the steady-state values of may be used to obtain in Eq. In obtaining Eq. we assumed that a Vg / az=avd az=o for the adiabatic area and (2 ) the vapor is not in contact with the pipe wall. Hence, this equation should not be used for separated flow. Equation allows interfacial drag to be computed from well-known pressure drop and void correlations. Other investigators take a somewhat different point of view and con clude that for bubble and intermittent flows the interfacial drag can be determined directly from drift flux parameters. Bestion writes the drift velocity, as
(3. 1 40). (3. 1 41)
B
(1)
1 52
UgI , Cm
(3. 1 42)
where = constant. For vertical flow, where buoyancy dominates, he expresses the interfacial friction force per unit volume, as
Fi , Fi a(1 - a)(PI - P g (3. 1 42) (3. 1 43) (PI-(3.P8g5)), PI -Coa)2 [ Vg - Clo (1 - a) ] 2 Fi a(1 -a)2C1m -(1 -Coa (3. 1 41) (3. 1 44) =
By means of Eq. elimination of
=
together with Eqs.
De
g)
and
V,
(3. 1 43)
we obtain, by
(3. 1 44)
Since Eq. is limited to only a few flow patterns in vertical flow, Eq. has a more general applicability. The early formulations of interfacial mass and heat transfer tended to be of the relaxation type in which the rate of interphase mass transfer de pended on the difference between actual and equilibrium values and some characteristic relaxation time. Kroeger 1 55 thus described the volumetric rate of vapor generation, r ' , during a pressure transient as
T H E RMAL A N A L Y S I S O F PWR S
318
r = P ili I
where
c
(ac / ap )s
tV cE
'T
cA
Pm
() ( ) ac dP s tV dt p a
(3. 145)
= vapor concentration {static qualitY = £Y/ [( PI / Pg ) - (£Y PI /a p)]} = change in c with respect to pressure based on maintenance of constant entropy = 0 if c E < cA or 1 + (C E - c) / T if C E CA = relaxation time = static quality at thermodynamic equilibrium = threshold value of CE up to which no vapor generation is assumed = mixture density, mass/vol.
>
More recent models tend to relate the mass transfer rate to interfacial heat transfer rates. This approach is used in the TRAC code. 102 TRAC as sumes that the steam-water interface is always at the saturation tempera ture and that the rate of interphase exchange of mass is defined by the energy exchange needed to maintain the saturation temperature. The vol umetric rate of energy exchange at the interface, q '7 (energy/ time vol), is then given by (3. 146) (3.1 47)
(ha)ig , (ha)i/
where = product of interfacial heat transfer coefficient and in terfacial area for gas and liquid, respectively, and Tg , TI , Tsa t = gas, liquid, and saturation temperatures respectively. TRAC provides separate corre lations for interfacial areas and heat transfer coefficients for each flow pat tern. Tables 6.IV and 6.V summarize correlations for calculating interfacial areas and interfacial heat and momentum transfer, which were developed for TRAC evaluation of LOCAs. It should be recognized that modeling of interfacial transfer rates re mains (1 995) an area of ongoing investigation and there is no general agree ment on the most appropriate models. 3 . 3 .4.5
III-Posed Equation Sets
The conservation equations for any of the previously described two-phase flow models can be written in the following general form: aY az
aY at
A - + B- - F = O
(3. 148)
H Y D R O D Y NAMICS
319
where A, B, and F are known functions of z , t, and Y . I t may be shown 1 S6 that the partial differential equations are total differential equations along the directions, dz/dt, obtained by solution of the determinental equation I A - B dz/dt l = o
(3. 1 49 )
The directions dz/dt so obtained are called the characteristic directions. When all of the characteristic directions are real, the problem is said to be well posed. When one or more of the characteristic directions are imaginary, the problem is said to be ill posed. The latter designation is used since the presence of an imaginary slope in problems describing propagation phe nomena is usually considered as indicative of a lack of physical reality. Some of the widely used two-fluid models are found to have complex characteristics. At the present time ( 1995), opinions still differ as to the correct interpretation of such complex characteristics and the proper course of action. Some workers have intimated that complex characteristics are necessary to the description of physical instabilities. However, Ramshaw and Trappl S7 have shown that systems with real characteristics can exhibit physical instabilities and that complex characteristics are not necessary to describe such behavior. They show that although stability and real char acteristics are not related in general, they become essentially equivalent at high frequencies. One might instead adopt the related position that equa tions having complex characteristics may correctly describe physical insta bilities. The analysis of Ramshaw and Trapp proves that although this might be the case at sufficiently long wavelength, the results become in creasingly erroneous as the wavelength is decreased and eventually, at very short wavelengths, equations having complex characteristics will predict unphysical instabilities, or, in other words, the problem is improperly posed. Krishnamurthy et al. 1 S8 have extensively examined the stability of the RELAP5 /MOD3 computer code, which is based on an ill-posed, one dimensional, two-fluid model. They found that model exhibited numeri cally convergent (stable) behavior within the discrete spatial bounds consistent with current (1992) usage. They concluded that a well-posed equation set is not necessary and that it is sufficient to require numerical stability only for the shortest waves of interest, i.e., A. = 2 (Ll z) where Llz is the shortest spatial increment to be used . 3 . 3 .5
Sonic Velocity and Critical Flow
A sudden change in fluid velocity causes a large pressure change that is seen as a pressure wave, across which there is discontinuity in pressure and velocity. The pressure wave, which can be either a compression wave (high pressure behind a wave front) or a rarefaction wave (low pressure behind a wave front), propagates at sonic velocity relative to the flow. It
T H E RMAL A N ALYS I S O F PWRS
320
can be set up by suddenly closing a valve in a flow line, by rupturing a high-pressure piping system (e.g., a LOCA), or by a steam burst in a reactor power excursion. The steam burst imparts kinetic energy, which is con verted into a pressure pulse when the flow is interrupted. Pressure waves of this general nature are often called steam hammer or water hammer. Pressure and velocity changes across an acoustic wave in one dimensional flow can be derived from the momentum equation
av av at ax +
V
= _� p
ap ax
(3.1 50)
x
where V is fluid velocity, P is pressure, and is distance. Since velocity change with respect to distance is relatively small in a straight pipe, the term is negligible, and Eq. (3.1 50) becomes
av Iax
aV at
t)
1 p
ap ax
t.
The pressure wave travels at the velocity of sound travel distance (a l in time Let
(3.151) al ;
therefore, it will (3.152)
where the negative sign is required for a wave moving in a direction op posite to V By substituting Eq. (3.152) into Eq. (3. 1 51 ), we obtain dV +a 1 de
- =
1
- -
dP
-
p de
(3.1 53)
By integrating, this yields P2 - P l V2 - VI
---=---'-
=
± pa l
(3.1 54)
Equation (3.1 54) shows that in subcooled water with a density of 62 Ib/ ft3 and a sonic velocity of 5000 ftl s, a 3 ftl s change in fluid velocity across the wave front causes a pressure pulse of 200 psi. When a one-dimensional acoustic wave travels in a fluid of equal and opposite velocity, the wave becomes stationary with respect to the earth. We then have critical flow at this point (fluid at sonic velocity), and down stream pressure signals can no longer be transmitted to the upstream fluid. Thus, the critical flow rate is limited by the upstream and critical point conditions, but not by downstream conditions. In the case of a rupture of a PWR piping system, the critical flow rate at the break determines the severity of the accident. In single-phase flow, sonic velocity a 1 and critical mass flow rate are directly and simply related: ai = gc v 2 / - (dv ldP )s '
(3.155)
321
H Y D RO D Y N A M I CS
where v is specific volume and subscript s indicates the derivative is eval uated at constant entropy. Critical mass flow rate G c is given by
(3.1 56)
Two-phase critical flow is, however, more complex: sonic velocity in a two-phase mixture varies with the amount of void present; velocity is usu ally measured at the wave front, where no phase change can take place instantaneously. On the other hand, critical flow rate is determined by the sonic velocity behind the wave front, where phase change does take place. This change depends on wave shape and bubble delay time which, in turn, depends on saturation pressure of the fluid. Therefore, it is necessary to consider sonic velocity and critical flows of two-phase mixtures separately.
3.3.5 . 1
Sonic Velocity
(1)
There are two limits for the response of a mixture of liquid and vapor to a pressure pulse: mass transfer between the phases maintaining thermo dynamic equilibrium, and (2) no mass transfer (frozen state) with liquid and vapor being independently isentropic. If no mass transfer is assumed, the computed velocity is often called the "frozen sonic velocity." If ther modynamic equilibrium is assumed, the computed velocity is substantially below frozen sonic velocity and the velocity actually observed at low and moderate pressures. If vapor and liquid can be considered a homogeneous mixture, sonic velocity can be quite simply derived from Eq. by using the mean density p, where
(3. 1 55)
Pg , PI aTP' aTP [ Pvapav (1 P-a)ia2l p] a,J I al al » a , (3. 1 58) v ai: [ a2 a(1 -a)Pl ] av Pv
a
(3.1 57)
where is void fraction and are vapor and liquid densities, respec tively. The frozen two-phase sonic velocity, can be written as: 159 -1 2
-
_
2
+
(3. 1 58)
where are sonic velocity in vapor and ,liquid, respectively. At low pressures where Eq. becomes =
+
(3. 1 59)
Dvornichenko 160 derived a slightly different expression for the frozen sonic velocity
(3. 1 60)
T H E RMAL A N A L YS I S O F PWRS
322
where X is quality. Grolmes and Fauske 159 found that Eq. (3. 158) fitted their low void fraction data very well. Therefore, it can be used to predict ve locity of the sonic wave front in a low-void bubbly flow in the absence of mass transfer. At high void fractions, the assumption of a homogeneous mixture is no longer valid, and increasing deviations from Eq. (3.1 58) are observed. In evaluating aTP by either Eq. (3.1 59) or Eq. (3.1 60), it is not obvious what path should be used to evaluate a7, . If adiabatic behavior is assumed, then a7, = gc P-Y / p u , where -y is the ratio of specific heat at constant pressure to that at constant volume. If isothermal conditions are assumed, a7, =
gc P/p,) .
Henry et al.161 studied the situation at higher void fractions where slip must be considered . They used p = 1 / [x v,, + 0 - x )vd in Eq. (3. 1 55) and ex pressed a2 in terms of liquid density PI ' gas density p,) ! and velocity slip ratio Equation (3. 1 55) becomes
S.
+ / (�;)
a ?p = [0 - X ) Pg XPd2
XPT
(:�)
+ 0 - X ) P� �t
(��) {:!) J
yt
(3. 161 ) + x(1 - X ) (PI - pg ) PIP' , where subscript s indicates the derivative is evaluated at constant entropy. When it is assumed that no phase change occurs, Eq. (3. 1 55) becomes
- (PI - PV )PIPg
(3.1 62)
Equation (3. 1 62) can be simplified for frozen annular flow (or stratified flow) as follows: For a stationary wave, Henry al.161 suggested
as � ( V aVI ) ap aTP ap ap =
For the vapor phase,
a
I'
_
et
(3. 1 63)
And, for the liquid phase, (3.1 64)
H Y D RO DYNAMICS
Therefore,
as = ( ) ap Pg-� X )Pg XP/F - X)(P/ - Pg)2} XPI X Pg2 -1 1 ai p
323
1
By substituting Eq. (3. 1 64) into Eq. (3.162), _ { [(1
2 aTP
-
+ x (1 ) 2 (1 + T a; a
+
(3.1 65)
For compressible flow at low pressures (al » a7.) ' sonic velocity can be ap proximated as aTP 2 (3.1 66) =1 + av or, in case of no slip,
( ) �X ( pg)PI 2 (-) 2 - --- a (-pg) P, a TP a"
_
1+
1
a
(3.1 67)
In high-quality or high void fraction flow, both Eqs. (3. 1 66) and (3.1 67) indicate that aTP � a v ' Henry al. 1 61 compared their air-water mixture data with Eqs. (3.1 58) and (3.155). For the bubbly-flow region, they found sonic velocity agrees approximately with the frozen model, Eq. (3.1 58); for stratified flow (non flowing system), Eq. (3. 1 65) was applicable. The stratified model of Eq. (3.1 65), which essentially gives the sonic ve locity of vapor, is applicable for high-void steam-water (annular) flow. This appears to be verified by steam-water data obtained by England al.} 62 Diech al.}63 and Dejong and Firey. l 64 Henry1 65 subsequently observed that the velocity of a wave front at low pressures and very low void fractions tends to approach that given by Eq. (3.158) (frozen model) with a; evaluated at isothermal conditions. For void fractions above those where Eq. (3. 1 58) strictly applies (a > 0.1), and below those of the annular region (a � 0.6), Henry 1 65 correlated the available data by (3. 1 68) aTP /aH = 1 .035 - 1 .671a
et
et
et
where aTP = actual propagation velocity in two-phase mixture and aH = propagation velocity determined from Eq. (3.1 58) with a�, evaluated under isothermal conditions. Equation (3.168) may simply be reflecting the difference in behavior be tween large and small bubbles. Hijikata al.'s166 study of shock waves in air-water mixture containing large and small bubbles showed that the two
et
324
TH E RMAL A N A L Y S I S O F PWR S
bubble sizes behaved differently. They concluded that the small bubbles had the same temperature as the liquid and moved with the liquid. How ever, larger bubbles were isentropically compressed and moved at a veloc ity different from the liquid behind the shock. As the void fraction is increased in a flowing system, the proportion of large bubbles is increased, thus leading to deviations from the homogeneous isothermal model. Despite the fact that the flow is probably not homogeneous under most conditions, Beattie1 67 finds that the two-phase sonic velocity may be ap proximated by an expression involving only flow quality and fluid specific heats. For those cases where and « aTP ' Beattie proposed
PI» Pg V
where
,,/1
=
x(c,)g + 0 - X)(Cp )' x(cv )g + 0 - x)(cv )r
(3.1 69)
and subscripts g and 1 indicate gas and liquid phases, respectively. Beattie was able to fit low-pressure air-water and steam-water data in the inter mediate void fraction range by assuming that qualities could be computed from measured void fractions using a drift flux model. The question of whether there is always negligible mass transfer at the wave front is not completely settled. On the basis of their data, Hamilton and Shrock 1 6R suggest that when the vapor bubble is very small, mass trans fer may not be negligible. Such a situation is likely in high-velocity dis persed flow where very small bubble sizes are present. Mass transfer rates may be enhanced at high temperatures where some investigators 169 indicate that the time required to nucleate new bubbles becomes very short. Hence, additional nucleation during the brief period of wave passage, which would enhance mass transfer and reduce the front velocity, might be possible. There is no agreement on this point. Direct measurements at high fluid velocity, high temperatures, and low or moderate values of a do not appear to be available. Measurements of sonic velocity in high-pressure (69-bar) steam-water flow in the dispersed annular region give acoustic velocities close to that of single-phase gas flows. This is in accord with the previously cited observations at low pressures. The effect of phase change slows down the pressure wave behind the front and makes it flatter than obtained in a fully frozen system (e.g., air water). Grolmes and Fauske 1 59 also observed this effect while studying the characteristics of both compression and rarefaction waves. Behind the wave front, the effects of heat and mass transfer become important and appear to prevent compression waves from steepening into shock waves, signifi-
H Y D R O DYNAMICS
325
cantly lengthening the shape of the rarefaction waves. This effect has led Henry165 to distinguish between the propagation velocity of a pressure pulse and the propagation velocity of a sound wave. Henry states that the frozen model holds strictly for a pressure pulse only. Propagation of a sound wave involves response to a continuous wave. At low frequencies, there is time for appreciable mass and heat transfer and, hence, lower prop agation velocities can be expected. High frequency waves can be expected to have propagation velocities that approach those of a pressure pulse. The detailed analysis of Ardron and Duffy1 70 is in accord with this point of view. They show that the speed of sound becomes a function of fre quency when thermal nonequilibrium effects are considered in a one component system. Ruggles et al.17 1 experimental data for the air-water system show that sonic velocity also varies with frequency in a two component system. However, in the nonvaporizing air-water system, the observed reduction in sonic velocity with decreasing frequency was con siderably smaller than would be seen in a vaporizing system. 3 .3 . 5 . 2
Critica l Flow in L ong Pipes
From the previous discussion, it is clear that there is an approach to ther modynamic equilibrium behind the wave front. Since critical flow is deter mined by conditions behind the front, some phase change must be considered. In a long pipe line, where there is adequate time for bubble nucleation and growth, thermodynamic equilibrium can be assumed. The homogeneous equilibrium model (HEM) is the simplest analytical model that can be postulated. The model assumes that 1 . both phases move at the same velocity the fluid is in thermal equilibrium 3. flow is isentropic and steady.
2.
G
By applying these assumptions to continuity and energy conservation equa tions, mass flow rate is found by where
G = [(2gcJ
)(Ho
-
H) /V2] 112 (G 2V2 ) / (2gcl ) XVg
(3. 1 70)
Ho = upstream reservoir enthalpy = H + H = local fluid enthalpy = Hf - X Hfg v = local fluid specific volume = ( 1 - X ) v r + = thermal energy to mechanical energy conversion factor.
J
For fixed stagnation (upstream reservoir) conditions, the critical mass flow rate is obtained by assuming the fluid entropy remains constant and finding the downstream pressure for which G exhibits a maximum. The results of these calculations for steam-water systems are presented in Fig. 3.20.
T H E RMAL A N A L Y S I S O F PWRS
326
1 00.0 �------'
1 0.0 L.L. W a:
�u X
e"
::> ..J L.L.
� ct � ..J ct
6.0
1 .0
� t
a: 0
0.1 PR E F hREF GR E F 0.01 0
Fig.
3.20
� � �
- 689.5 _
=
=
2.0
1 00 I bf/in. 2
k N /m 2
( 2.326)
�
1 00 Btu/Ib m 1 05 J /kg
4882
1 000 I b m /ft 2
kg/5·m2
�
4.0
� 6.0
8.0
1 0.0
1 2.0
Critical mass flux and homogeneous equilibrium, steam-water mixture [reproduced from NEDO-21052, General Electric Co. (Sep. 1 975)] .
Moody 1 72 considered! the available data for blowdown in long pipes. From an examination of nonequilibrium behavior during depressurization, Moody concludes that if pipe length is over five inches, equilibrium states can be expected. He found that available data on the blowdown of reser voirs through pipes of lengths 5 in. were predicted by Fig. 3.20 when calculations were based on reservoir conditions. This is in agreement with Edwards ' 1 69 observations. Moodyl 72 points out that although homogeneous theory using reservoir conditions provides a good prediction of critical flow rates, it provides a
>
H Y D RO D Y NAMICS
poor prediction of pressure at the exit of the blowdown pipe. He concludes that mass fluxes are limited by homogeneous choking near the pipe en trance, but that a transition to slip flow occurs before reaching the exit. A second choked condition near the exit is produced and, if critical flow rates are to be evaluated on the basis of local conditions near the pipe exit, a slip flow model must be used. Critical flow models for use with exit conditions, and based on high slip ratios at the exit, were developed by Moody, I 72,1 73 Fauske, 1 74 Zivi, 175 Cruver and Moulton,1 76 and others. A major difficulty with these models is that the slip ratios used are very much higher than those observed in high ve locity flow under nonchoked conditions. In addition, it has been shownI 48, 1 77, 1 78 that the unrealistically high slip ratios assumed by these models lead to imaginary characteristics. Hence, they render the system of conservation equations ill-posed for much of the domain of interest. Tentner and Weisman l 48 showed that, when realistic slip ratios were used together with the single-fluid conservation equations of Eq. (3.1 29) through (3.131), real characteristics were always obtained. They found that the computed critical flow rates were very close to those of the HEM model. It is now generally agreed that thermal nonequilibrium effects account for the apparently anomalous behavior at the break. Experimental data show a very sharp pressure gradient at the exit. 1 79 Such a gradient would be expected to produce nonequilibrium with void fractions and quality which are lower than predicted by equilibrium theory. Thus, if discharge predictions are to be made on the basis of break conditions, a nonequili brium theory should be used. Nonequilibrium effects also need to be considered in some instances where reservoir conditions are used to compute the discharge flow rate. If the liquid in the upstream reservoir is sufficiently subcooled so that very low qualities are observed at the break, discharge flow rates are found to be significantly higher than predicted by the HEM. Henryl80 devised a simple model for pipes of intermediate lengths where thermodynamic equilibrium is not attained. He described the nonequili brium behavior in terms of the parameter N, which relates the actual change in quality with pressure, dx/dP, to dXe / dP, the rate of change in quality occurring under equilibrium conditions: dx/dP = N (dxc / dP )
(3. 1 71 )
On the basis of experimental data, Henry concluded that the actual exit quality, at the break of a long tube was given by
XLT '
(3. 172)
where Xc = thermodynamic equilibrium quality at exit and
<xc <
0.05 N = 20x!, a N=1 0.05 � Xc .
T H E RMAL A N A L Y S I S O F PWRS
328
For tubes having a length to diameter ratio (LI D) of less than 12, he assumed vapor generation could be neglected. For longer tubes, he as sumed that the actual exit quality xex exponentially approached the long tube value. That is
Xex = xLT { l - exp[
B A
-
B
(LI D -
A)]}
(3. 1 73)
where = 12 for tubes with sharp entrance and 0 for a smooth inlet and = 0.0523. With the assumption of isothermal behavior, the critical mass flow is calculated, assuming :slip is unity, as
G; =
[ XUv
]'
1
(3.1 74)
dXe - - (uv -v )NP dP ex I
where Vv and VI are the specific volumes of vapor and liquid and the sub script ex indicate that all quantities are evaluated at the exit conditions. To do so, the exit pressure Pex is required. This is obtained from TI
P G V = peox = 1 - p; 2KIO2 + xex (vg,ex - V1,0 ) 0
(3.1 75)
where Kc is the contraction coefficient at the pipe exit. The subscript 0 in dicates reservoir conditions. By iterative application of Eqs. (3. 1 73), (3. 1 74), may be evaluated. and (3. 1 75), Nonequilibrium effects are also encountered in the discharge from long pipes when annular flow at very high void fractions exists at the exit. At the high velocities seen in choked flow, most of the liquid carried by the fluid is in the form of entrained droplets. The critical velocity is close to the critical velocity of the vapor flowing alone. Since the droplets move at a velocity close to the sonic velocity of the vapor, the mass flux discharge is significantly higher than HEM predictions. Grolmes et aU8 1 have pro posed a simple correlation that merges the high-quality flashing flow data with the equilibrium model ,at low qualities. The correlation is stated to apply over the entire quality range, c
Gc
{ [ Gc = l +
x
ob
0.6(1 - lnP* Ib)
] �} 1!2 ( 2
mwcp
Vfg y'cp To
where
Gc = critical mass flux, kg/m2 s Xo = reservoir qUlality b
R= T=
=
13 for water, 10.6 fOir most other substances ideal gas constant, 8314 J Ikg mol temperature,
K
K
Hfg
)
(3.1 76)
H Y D R O DY NAMICS
329
specific heat of liquid, J /kg K mw = molecular weight of vapor P* = exp[b(1 - 1 / T * )] T * = / (temperature at normal boiling point at atm pressure) . Hfg = enthalpy change on vaporization, J / kg 1Jfg = change in specific volume at vaporization, m3/ kg. Cp =
T
Note that Eq. (3.1 76) is written in a manner which allows it to be applied to a variety of fluids. 3.3.5 . 3
Critica l Flow i n Short Pipes, Nozzles, a n d Orifices
When a reservoir discharges through a short passage a high degree of ther mal non equilibrium is encountered. If the length of the flow passage is negligibly small (e.g., discharge across an orifice), vapor formation has little influence and the discharge mass flux, G, may be calculated by the orifice equation181 (3. 1 77) where
Pm
Cd
Po , Pa
gc
= =
density of fluid mixture upstream of orifice orifice discharge coefficient (0.61 at high Reynolds numbers of discharge flow) upstream and ambient pressures, respectively gravitational conversion constant (1 .0 in 51 units).
At very high pressure differentials, true choking can be encountered. The flow is then limited to a mass flux equal to the product of the mixture density and frozen sonic velocity over the area of the vena contracta. Both mass flux and sonic velocity are evaluated at the pressure upstream of the orifice. Vaporization can influence the discharge of a reservoir through a short pipe or nozzle. A commonly encountered condition is one in which there is subcooled fluid just upstream of the break. Under these conditions, two types of choking mechanisms were seen by Zaloudek. 182 Zaloudek observed tha t an upstream choke can form at a vena contracta and a downstream choke can form as the back pressure is built up at the exit edge by flashing. He found that when choking occurred at the vena contracta, critical mass velocity was given by Eq. (3.1 77) with Pa replaced by the saturation pres sure and is an empirically determined constant found to range from �0.61 to 0.64. When downstream choking occurred, the critical flow rate agreed with Burnell's 1 83 correlation. Burnell 183 previously recognized the existence of a metastable state in the flow of flashing water through nozzles and hypothesized that water
Cd
T H E RMAL A N A L Y S I S O F PWRS
330
0.30 0 . 28 0 . 26 u
0.24 0.22 0.20 0. 1 8 200
0
Fig.
3.21
400
800 600 PSat l p s i a
1 000
Pressure coefficient for Burnell critical flow equation [from Ref. 1 83].
surface tension retarded the formation of vapor bubbles, thus causing the water to be superheated. He developed a semi-empirical method for pre dicting the flow of flashing water through short nozzles and correlated the discharge mass flow rate as Gcrit =
{2gcPI [P
u pstream
-
(1 - C ) Psat ]}l12
(3.1 78)
where C was related to bubble delay time, which is a function of surface tension. The value of C that is recommended for design use is given in Fig. 3.21 . The value of C determines the pressure undershot at the exit due to superheating of the liquid. Sozzi and Sutherland 184 also observed nonequilibrium effects in blow down through short nozzles. They saw a strong dependence on nozzle length with the shorter nozzles giving higher flow rates. They found that fluid passing through a very short length will not have sufficient time to completely nucleate before leaving the tube. For nozzle lengths of about five inches, agreement between observations and homogeneous predictions was obtained. They also observed that critical mass flux decreased with increasing throat diameter. Abuaf al. 185 provide a more fundamental analysis of flashing of sub cooled liquid across a nozzle. They replaced the term (1 c) Psat in Eq. (3.1 78) and wrote
et
-
Gc =
[2gc P[(P
ups tream
-
Psat + D. PF ) f/2
(3.1 79)
where D. PF = pressure difference corresponding to the maximum superheat which can be sustained without flashing. The value of D. PF is based on a
H Y D R O D Y NAMICS
modification of Alamgir and Lienhard ' s 1 86 static depressurization super heat, which allows for turbulence effects and change in pressure with po sition. It is given by
(3. 1 80) where
2.
= apat V apaz = G;PT d (InA) dz _
I
+ 2. '
and
= ib = = �Po = Tc = Tr = Tsat /Tc = 2. '
A
k
IT
nozzle flow area reservoir depressurization rate 0 turbulent intensity Boltzmann' s constant pressure undershoot in absence of turbulence effects critical temperature
= dP / dt
surface tension.
(3. 1 78)]
While this approach correlates available data and provides a more mech anistic view of nozzle behavior, the Burnell equation [Eq. appears to be adequate for design purposes. Henry and Fauske 187 developed a model for critical flow in nozzles and short tubes which allows for nonequilibrium effects and considers a two phase mixture upstream of the break. They proceed in a manner similar to that used by Fauske, 174 but do not assume thermodynamic equilibrium be tween phases. They use the empirical parameter to relate actual to that under equilibrium conditions in accordance with Eq. The experimental parameter is given by
(dxE /dP)
N N=x N = 1.e0/O. 1 4 dXE G = [ XnPO Vv v ) { O5v-Xo)N e dP
for for
dx(3./ dP1 71).
xxee �0. >0. 11 44 Xocp[O /n)- O /'Y)] } ] P(5vo
They concluded that the critical mass flow rate was given by 2
C
--
- (v - v/0
- 51e
----!....-
- - -
5 /0 )
(3. 1 81)
T H E RMAL A N A L Y S I S OF PWRS
where Xo
=
n =
'Y
= =
(Ie '
S ic =
s"o ,
S IO =
quality at stagnation conditions polytropic exponent for vapor compression isentropic exponent for vapor compression = / c" specific heats of vapor at constant pressure and volume, respectively entropy under equilibrium conditions of vapor and liquid, respectively entropy at stagnation conditions of vapor and liquid, respectively. )
Equation (3.181) is coupled with the momentum equation describing the pressure history to obtain a solution in terms of stagnation conditions ( l - Xo ) V/O ( PO - PI ) +
Xo'Y _ [ ( l - X o ) v/ + Xo vv, F 2 , ( POV"O - PI VI)I ) G( 'Y - 1 2
--
(3. 1 82)
where the subscript t indicates throat conditions. By substituting this equation into Eq. (3.181) and by rearranging, we get the ratio of throat pressure, PI ' to stagnation pressure, Po . (3. 1 83) where
and subscript t indicates throat conditions. Once 11 is obtained by solving the transcendental equation, Gc is obtained from Eq. (3. 182). Note that the correlation given for holds only for a dispersed mixture of vapor and liquid. In the discharge of a saturated or subcooled liquid through an orifice or short tube, a separated flow pattern is often observed .
N
H Y D R O D Y N AM ICS
TA B L E 3 .VI I I Applicability o f S i m ple Models for Calculation o f C ritical Flow Rate
Apparatus Long pipe Long and intermediate length pipe Long pipe Intermediate length pipe Nozzle Short pipe Nozzle or short pipe Orifice
Note:
Upstream Conditions Two phase (a < an ) to moderately subcooled liquid Two phase (a > all )
Applicable Calculation Procedure HEM model (Fig. 3.20) Grolmes et al. [Eq. (3. 1 76)]
Highly subcooled liquid Subcooled liquid to twophase with a < a" Saturated or subcooled liquid Saturated or subcooled liquid
Henry lEq. (3. 1 74)] Henry [Eq. (3.174)]
Saturated or subcooled liquid Two-phase mixture
orifice equation [Eq. (3.1 77)]
Two-phase mixture (a < an )
all = void fraction at which annular flow begins.
Burnell [Eq. (3. 1 78)] Burnell [Eq. (3.178)] or orifice equation if back pressure builds up Henry and Fauske [Eq. (3. 1 81 ) ]
minimum of value from Eq. (3. 1 77) or mass flux based on frozen sonic velocity
The proposed correlation for N is not applicable under these conditions and Burnell' s equation should be used for Gc prediction. The solution of the transcendental equation for T) is time consuming. Because of this, when this approach is used as part of an accident analysis computer program, it is often represented by a set of polynomial equations that have been fit to the results over the parameter range of interest. 188 The applicable regions of the various simple critical flow models are summarized in Table 3.VIII. 3 . 3 . 5 .4
Critical Flow A cross Safety and Pressure Relief Valves
Safety and pressure relief valves are designed for the release of vapor. Section III of the American Society of Mechanical Engineers (ASME) Boiler and Pres sure Vessel Code considers such valves to behave like an orifice having a flow area, equal to the circumference of the valve seat, TID, times, the valve lift (distance valve seat moves on opening). At the high-pressure difference expected in nuclear vessel operation, the flow is assumed to be choked. The design equations recommended by the ASME may be written as 1 89
Ao ,
L,
(3. 1 84)
T H E RMAL A N A L Y S I S OF PWRS
3 34
where Ao = (-rr D )L W
ell
Gc
= = =
mass flow ra te, mass 1 time valve coefficient correcting for non idealities theoretical critical mass flux through nozzle of throa t area Ao .
The valve of Gc is based on the sonic velocity of steam and is determined by (3.1 85) where
P = o
upstream pressure ideal gas constant Z compressibility factor = cJ.j c" = 1 .1 3 for steam n = + 1 ) / ( "1 - 1 ) To = upstream temperature.
R "I ( "I
Sallet 1 89 observed that the el> values used in Eq. (3. 1 84) are based on the assumption that the back pressure is atmospheric pressure. This is not the case in a LOCA. He notes that when the back pressure increases signifi cantly the value of e" is reduced. His data show increasing the back pres sure to 50% of the discharge pressure decreases ev by about 13%. When a liquid or low-void fraction two-phase mixture is discharged through a safety or relief valve, the sonic velocity is higher than for all vapor flow and one would expect that the nonchoked orifice discharge equation would apply [Eq. (3. 1 77)]. In Sallet' s 1 89 study of the blowdown of refrigerant liquid, the discharge mass velocity was predicted by the orifice equation, with flow area of Ao and back pressure at that of the downstream tank, when Po was significantly above the set pressure. However, the dis charge rate fell significantly below these predictions as Po approached the set pressure. Orifice-like behavior when discharging liquid was also observed by Chen They examined the behavior of a spring-loaded safety valve with a short throat. Their tests examined the discharge of refrigerant 1 1 3 with both liquid and two-phase mixtures upstream of the valve. At the relatively low pressure differentials (up t@ 4.5 bar) of their tests, the valve acted as an orifice with a variable area. Their results could be correlated by
et al. 190
W/ (
A P )1i2 K l =p set
gcPm u
( p0 - -PmVI� ) gc P
set +
+ K2 ,
(3. 1 86)
H Y D R O D Y NAMICS
where
VmW Pm
= = =
Kl
=
K2
=
Pset
=
Po
discharge mass flow rate, mass/time mixture velocity based on upstream conditions mixture density based on upstream conditions constant for given valve indicating change in flow area produced by additional force on valve seat constant for given valve indicating flow area provided by initial valve lift set or relief pressure system pressure.
The value of the right side of Eq. (3.186) reaches a maximum value when the valve is fully open. It then acts as an orifice of constant area. If much higher pressure differentials had been used, a choked flow condition would be expected when the mixture velocity reached the frozen sonic velocity. The Chen af. 190 study explains the reduced flows observed by Sallet at low pressures. As the upstream pressure approached the set pressure, the valve acted as an orifice with a variable area. The results of CISE tests 191 of safety valves discharging (60- to 1 20-bar) water and water-steam mixtures were entirely different. The valves, which had relatively long throats (80 mm < L < 1 1 6 mm) showed choking in the vicinity of the valve exit for all test conditions. With the flow taken as that of the valve throat, critical flow rates were closely matched when HEM results were multiplied by a coefficient slightly below unity. This result agrees with the recommended procedure for sizing safety relief valves in the chemical process industry. 192 It would seem that tests of the specific valve geometry used are needed to establish definitively whether the valve behaves as an orifice with an area determined by the valve lift or if the valve will exhibit choking at the exit of the throat.
et
3 .3 .5 .5
Comprehensive Critica l Flow Models
All of the previously described critical flow models make particular as sumptions regarding the flow behavior and ignore some of the conservation equations. Many of the more recent attempts to determine critical flow rates have been based on models that consider all of the conservation equations and thermodynamic nonequilibrium. The models generally allow for delay in the initial vaporization of subcooled liquid and a finite rate of bubble growth or heat transfer between phases once bubble nucleation has oc curred. The resulting equations are then numerically integrated in a step wise fashion along the flow passage. When critical flow actually occurs, a pressure discontinuity exists across the standing wave at the flow-passage exit. Most of the models consider that critical flow has occurred when the assumed mass flux leads to an axial pressure gradient at the exit which is
336
T H E RMAL A N A LY S I S O F PWR S
equal to or above some arbitrary high value. Models of this nature have been developed by Ardron, 19J Dobran, I 94 Elias and Chambre, 1 95 Dagen ct al} 96 and others. Dagan et af. 196 use five conservation equations that include mass and momentum equations for each phase and a combined energy equation. A sixth equation is added describing the growth of vapor bubbles along the flow path and a seventh equation relates time to the axial position along the pipe. The seven variables described by these equations are liquid tem perature, gas velocity, liquid velocity, bubble radius, flow quality, pressure, and time. The Dagan et al. 196 model is self-consistent and it may be used all along the discharge line. It also was able to predict the results of the large pipe experiments at Marviken. 1 97 The ability to predict large-pipe data is signif icant since such data represent behavior in a large loss of coolant accident. Many models do not provide very satisfactory predictions of these data. Instead of using the exit pressure gradient as a criterion for critical flow, other comprehensive models have used the characteristic slopes. Recall (see Sec. 3.3.4.5) that the solution tOf the determinental equation, Eq. (3.149), for the slopes (dz/dt) provides the characteristic directions. These characteristic directions represent propagation velocities. Since at the critical condition information from downstream cannot be propagated upstream, we require that the lowest value of dz/dt be zero when critical flow is attained
]
min[ (dz/dt ) i = 0
for i = 1, 2,
.
,n
(3. 1 87)
where n is the number of equations defining the A and B matrices. In most cases, analytic solutions for (dz/dt) cannot be found and numerical solutions are required. Note that unless the nonequilibrium terms in the defining equations are associated with a derivative term, they do not affect the val ues of the characteristic directions. One of the simpler models based on the method of characteristics is that due to Kroeger. I 55 He used a five-equation drift flux model in which the vapor phase was assumed saturated and local pressure was used. Kroeger was able to solve the resultant characteristic equations and found for the case of zero drift velocity (dzjd t \.
=
Vm ± a H F
(dz/dt)2,3
- (-
=
C_ � J Z J3a �
V,
)
(3.1 88) (3.189)
where Vm is the mixture velocity. Hence, at critical conditions: V11/. - a HF
1
1 Z I f3pa� F
(3.190)
H Y D R O D Y NAMICS
aHF } au d 1 / / H du x -U -x -' -(1-x) aHF = u{ -Cpl (-aTI) [ ( dPe.:) J (dUe.:dP ) ( aP ) (3.1 91) U= Z'= ![ (Ux -U/) + HxCp-l HI ( aulap) J (3.1 92)
where
is the homogeneous frozen sound speed given by
1/2
' -
p
where
sa t
T
sat
mixture specific volume and Z ' is given by 1)
p
dX£ ( 1 + Xc -x)
The factor � accounts for nonequilibrium mass transfer:
dP
�=-
Xc = = 1. 1 10 - 3
(3. 1 93)
T'
--
=
where equilibrium thermodynamic quality corresponding to the local mixture enthalpy, relaxation time (Amos and Schrock 1 99 recommend ' s), and X T ,
dxc (1 - X) (d) +x (:�x)
dP
sat
Htg
sat
(3. 1 94)
(3. 1 94)
Note that the conversion of thermal energy into kinetic energy has been neglected in Eq. and the static mixture enthalpy is taken as constant. It should also be noted that the characteristic directions can be determined without setting the drift velocity to zero. However, a numerical solution is then required. The assumption of zero slip at the choke can be justified on the basis of the previously cited studies showing that the slip ratio must be close to in order to obtain real characteristics. Use of the characteristic directions for determination of critical flow rates requires that the critical flow calculations be coupled to a model that can compute the local conditions at the choke location. Shrock and cowork ers 198, 199 have shown that, when used with their fluid modeling, Eq. can satisfactorily predict the subcooled blowdown behavior seen in the large-pipe experiments at Marviken. 197 Schrock et based their computa tions on the use of mixture conservation equations written in terms of the mixture velocity, liquid temperature, pressure, and quality, The conservation equations employed were
1.0
V, Mass:
al.
TI,
P, dPdz V1 -dVdz --U1 [ (1 - x) (-iapJuI) +x (-dUgdP ) J -+dx ( 1-x) (-du/ ) -=0 dTI +--Ug-U-UI- -+ dz aTI dz sa
TI
p
(3.1 90) x.
t
(3. 1 95)
T H E RMAL A N A L Y S I S OF PWRS
338
(3.1 96) Momentum: dPdz + � dVdz = (dPdz ) + £ Energy: [ (1 _ X) (:�I) TI + x (d:;)J�� v�� + (Hg - HI ) �: =0 (3. 1 97) (dP/dz)! _
!
u
u
+
where is the frictional pressure drop, u, ug , U/ are specific volumes of mixture, gas, and liquid, respectively, and other symbols have their pre vious meaning. For blowdown conditions, the vapor is taken as at satura tion and vapor properties evaluated from an appropriate equation of state. Properties of the superheated liquid were estimated by extrapolating sub cooled properties to the superheated region. Since only mixture velocities are considered, this implies tha t slip is taken as negligible throughout the blowdown line. The liquid superheat was correlated by
dP-- (-dP) -=dP dT b -t IT , ) dz dT dz tr= fzrz Vdz(z) dPb T' (3. 1 93). dPb dPb =O.013a'p1 .38 a' P exp(
sa t
where
r
(3. 1 98a)
is the residence time of two phase mixture in duct, is the pressure undershoot at inception of flashing, and is the relaxation time as used in Eq. Lee and Schrock198 provide a correlation for that leads to better results than obtained with earlier estimates of this quantity. For saturated liquids, they obtain improved results by using the empirical expression
(3.1 98b)
where (surface tension) and are evaluated at inlet conditions and all quantities are in SI units. The ability of the approach of Schrock and coworkers to predict sub cooled blowdown data confirms the importance of thermal nonequilibrium and the minor influence of slip on this behavior. Determination of critical flow conditions via characteristic directions de termined by a full two-fluid model has been accomplished by Lahey,200 Sami and Duong,201 and Micaelli al. 202 Lahey'S modeJ2°o [Eqs. and does not contain any heat or mass transfer terms as sociated with the spacial or time derivations. The characteristic directions obtained from his model therefore do not reflect the effect of interphase mass transfer. Sami and Duong20 1 introduce derivative terms into the mass transfer rate. However, the formulation of these terms is such that they only appear
(3. 1 35), (3.1 36)]
et
(3. 1 32),
H Y D RO D Y N AMICS
339
in the characteristic equations as a multiplier for the difference between the gas and liquid velocities. Since this difference approaches zero at the high velocities characteristic of critical flow, nonequilibrium mass transfer will have little direct effect on the results. However, the use of these nonequi librium models in the line leading to the break will produce qualities below the equilibrium quality at the choke point. Therefore, use of such two-fluid models will significantly affect the predicted break flow even though the characteristics do not themselves reflect nonequilibrium effects. Integration of any of the comprehensive models into one of the computer programs used for PWR accident analysis is difficult. The accurate deter mination of break conditions requires the use of much shorter length steps in the break line than those commonly used. In the explicit and semi implicit solution schemes of most programs, reduction of the length step anywhere reduces the allowable time step and hence increases solution time. The CATHARE program202 avoids this difficulty. This code uses a fully implicit solution scheme that allows the use of a very fine mesh near the break without a reduction in the time step. CATHARE determines the critical flow rate from the characteristic equations of a six equation two fluid model. This requires the solution of a sixth-degree polynomial equation. While many systems analysis programs avoid lengthy solution times by using very simple critical flow models, others (e.g., RETRANJ8) use ap proximations of more complex analytical models. This is done by devel oping an algebraic equation that fits the solutions obtained from the complex model of interest over the range of conditions expected in a LOCA. 3.3.5 .6
Oecay of Pressure Wa ves
When a pressure wave reaches a rigid dead end, it is reflected as a wave of equal magnitude and sign; that is, a rarefaction wave is reflected as a rarefaction wave and a compression wave as a compression wave. When a wave reaches an open end or large reservoir, it is reflected as a wave of equal magnitude but of opposite sign. When a wave encounters a change in area, a portion of the wave is transmitted in the original direction of travel and a portion is reflected back. With the assumption of one dimensional streamline flow, it can be shown203 that the amplitude of the transmitted wave is given by Ll P tra nsmitted Ll P incident
2Aincident Aincident
+
Atra nsmi ssion
(3. 1 99)
and the ratio of reflected Ll P to the incident Ll P is Ll Preflected Pin cident
�
Ain c ident - Areflecti on Aincident Areflection
This behavior is illustrated in Fig. 3.22.
+
(3.200)
T H E RMAL A N A L YS I S O F PWRS
340
P = P2 A = l nc i dent
Time 1
A
P = P2 A = A I n c i dent ,..--_--"''''--__ �
Time 2
a
A = A
Fig.
3.22
T ra n s m i
Transmission and reflection of acoustic wave at area change.
In a frictionless system, the combination of forward and reflected waves is undamped and sets up a standing wave. In any real system, friction at the walls gradually reduces the amplitude of the waves. In a straight pipe, Lieberman and Brown203 give decay with time t as il P = (il Po )
(3.201)
exp( - I Vtl D)
where diameter of the pipe I = Fanning friction factor V = fluid velocity il Po = il P at zero time. D
=
20
1
Ilu 1(i)
0.005,
For a pipe diameter of ft, a Fanning friction factor of and water velocity of ftl s, time constant D is s; thus, the frictional effect is relatively small. When a pressure wave is reflected by a body that can significantly de flect, the amplitude of the reflected wave is appreciably reduced. The reader is referred to Streeter and Wylie204 for discussion of this case. In a reactor system, the piping and major components deflect very little and can usually be considered as rigid; however, deflection of any thin wall internal struc ture should be considered. The many area changes in a reactor loop lead to the appreciable atten uation of an acoustic wave as it proceeds through the system. However,
H Y D R O D Y NAMICS
341
the reflected waves set up at the area changes and the subsequent trans mission and reflection of these reflected waves require an analysis that can follow this complex behavior in its entirety. The transmitted and reflected waves can act to reinforce each other in a particular region and to produce a high pressure across a particular component. The arithmetic method204 is conceptually the simplest procedure for es tabli shing system behavior. The method neglects friction and generally as sumes a horizontal system. The procedure basically requires that each wave be followed through the system. Pressure change across each wave is de termined by using Eq. When reflected and transmitted waves are created at area changes, their subsequent behavior must be followed. The pressure-time history at a given location is determined by summing the pressure changes that occur as the various waves pass through the location in question. Tediously tracking individual waves makes the arithmetic method un suitable for all but the simplest of systems. The method of characteristics204 is most commonly used for solution of waterhammer problems. For single phase flow, the continuity equation is written as
(3.1 54).
(aH/at) (a /g)(av/ax) V(aH/ax) 0 g (aH/ ax) V(aV/ ax) (aV/ at) (fV/2D)I V I 0 +
i
+
=
the momentum equation is written as +
+
+
=
(3.202) (3.203)
where
D x a1 V
H
= = = = =
fluid head pipe diameter axial distance sonic velocity fluid velocity.
These partial differential equations are then converted into total differential equations that are subsequently expressed in finite difference form. The system is then divided into a series of control volumes and a set of finite difference equations is written for each control volume. Details of the nu merical solution procedure are described in Sec. The simplest system modeling that can be used in this analysis is one that considers the reactor system to be one dimensional and, thus, to consist of a series of linear segments. While this is a reasonable way to simulate piping segments, it is not a good model of multidimensional regions within the reactor vessel. Streeter and Wylie20S pointed out that a more realistic modeling of such regions can be obtained by using an equivalent intercon nected piping network instead of a simple linear arrangement. Fabic206 sug gested further that when such network modeling is used, sonic velocities
4.1 .2.3.
342
T H E RMAL A N A L Y S I S O F PWRS
should be adjusted for the fact that piping network paths are longer than those actually traveled by the sonic waves. He proposed that the sonic velocities be multiplied by V2 and y3, respectively, for each two- and three-dimensional region. Comparison of the calculated and experimental results shows improved agreement when the equivalent piping network with modified sonic velocities is used. 207 The pressures produced by wave propagation within a PWR primary coolant system are of greatest concern in the brief period while the system is still subcooled after a hypothetical LOCA. Once boiling begins, sonic velocity is greatly reduced and the pressure waves are damped out. The WHAM code208 has been widely used for analyzing pressure wave behav ior in subcooled reactor systems. 3 . 3 .6
Pump Behavior with Two-Phase Flow
Two-phase fluid within a centrifugal pump leads to substantial degradation of performance over a wide range of void fractions. Two different ap proaches have been used to treat this problem. In one approach, analytic models are used to predict two-phase pump performance209 , 210 from single phase performance. Model constants are adjusted to conform as well as possible to experimental data. Due to the complexity of the problem, this approach has been only moderately successful. The second approach uses experimental data more directly via homol ogous curves for degraded operation. Extensive tests of centrifugal pumps similar to those used in PWRs have been carried out in the semiscale system at Idaho National Engineering Laboratory. 21 1 These tests indicate that for between about and there is little degradation in performance. The performance then falls sharply and remains nearly constant until Performance in this region is said to be fully degraded. Performance then rises sharply as increases above and then performance remains nearly undegraded. Based on this information, the RELAP4 and TRAC codes 102 contain equations that fit semiscale pump homologous curves for head and torque in the fully degraded region. Performance is considered undegraded until voids reach about Over a narrow region, the pump head and torque are brought to the fully degraded level via a linear inter polation between the degraded and undegraded performance curves. The performance then remains fully degraded until voids exceed approximately where an interpolation scheme brings the performance to undegraded levels.
a
0 20% a
80%
a =80%.
7%.
80% 3 .4
F LOW I N STAB I L ITY
The term flow instability refers to flow oscillations of constant or variable amplitude that are analogous to vibrations in a mechanical system. Mass flow rate, pressure drop, and voids can be considered equivalent to the
H Y D RO D Y NAMICS
343
mass, exciting force, and spring of the mechanical system. In this connec tion, the relationship between flow rate and pressure drop plays an im portant role. Flow oscillations can be aggravated when there is thermohydrodynamic coupling between heat transfer, void, flow pattern, and flow rate; however, oscillations can occur even when the heat source is held constant. Both hydrodynamic and thermohydrodynamic instabilities are discussed, but the more elaborate nuclear thermohydrodynamic insta bilities of boiling channels in water-cooled reactors are beyond the scope of this book. Flow oscillations are undesirable in boiling, condensing, and other two phase flow devices for several reasons: 1 . Sustained flow oscillations can cause undesirable forced mechanical vibration of components. Flow oscillations can cause system control problems of particular im portance in liquid-cooled nuclear reactors where the coolant (such as water) also acts as a moderator. Flow oscillations affect local heat transfer characteristics and boiling crisis.
2. 3.
Thus, the critical heat flux (CHF) was found by Ruddick212 to be reduced when the flow was oscillating. This adverse effect was also found by Lowdermilk et a1.2 1 3 Mayinger et a1.214 found a similar reduction of CHF in an oscillating water flow at psia.
40%
3.4. 1 3.4. 1 . 1
2000
Natu re of Various Flow I nstabil ities Flow Pattern Instability
This occurs when flow conditions are close to the point of transition be tween churn flow and annular flow. A temporary increase in bubble pop ulation in churn flow (arising from a temporary reduction in flow rate) can change the flow pattern to annular flow with its characteristically lower pressure drop. When the driving pressure drop over the channel remains constant, flow rate attains a greater value. However, as flow rate increases, the vapor generated (even for unchanged heat transfer characteristics) can become insufficient to maintain annular flow. The flow pattern then reverts to that of churn flow and the cycle is repeated. The low-pressure drop chara cteristics of annular flow have been experimentally demonstrated by Wallis.215 The cyclic behavior is partly due to the delay incurred in accel eratio n and deceleration of the flow. 3 .4.1 .2
Ledinegg Instability
Consider the operation of a boiler tube to which there is a constant heat input. At low flow, the boiler exit quality and velocity are high since the fluid is entirely, or almost entirely, vapor. This high vapor velocity accounts
T H E RMAL A N A L Y S I S OF PWR S
3 44
for much of the pressure drop. When the flow is increased, slightly more vapor is generated and the pressure drop increases. However, as flow in creases further, exit quality begins to decrease and velocity at the tube exit decreases. The pressure drop decreases and continues to decrease until the tube contents are all water. The pressure drop then increases as the water velocity increases. Experimental observations of this phenomenon are shown in Fig. (Ref.
3.23
216).
75 �-------r--'
"-
50
2 0
'0 �
II>
Q) ..L: U C
a:
0 �
0
�Q)
ct
25
Out l et D-C Heat F l ux, Pressure, I n l et In l et 2 Run psia Temp, o f Btu/(h ft ) Su bcoo l i ng , + 242 1 3 9 000 215 1 46 • 23 8 1 50 249 000 215
a
Fig.
3.23
5 I n l e t Ve l o c i ty, fps
of
10
Static pressure drop through a O. 1 74-in.-i.d. stainless-steel heated single tube (from Ref. 216).
H Y D R O DYNAMICS
345
B O I L E R I N LET - - - FEE D S Y S TEM EXIT
---
POSITIVE
N EGATIVE
N EGAT I V E
STA B L E
U N S TA B L E
STA B LE
\
PRES SURE
W E I G HT F L OW
Fig.
3 .24
-
\
��\
Boiler resistance (from Ref. 217).
3.24,
The effect of this phenomenon on stability can be understood by refer ence to Fig. where we consider the operation of a boiling channel with a constant exit pressure. A portion of the pressure versus flow curve, which occurs at very low flow rates, is shown at the left. The corresponding curve for the boiler feed system is also shown. The system will operate where the two curves intersect. If flow to the boiler increases by a small amount, d boiler pressure becomes greater than supply pressure. This decelerates the flow and the system returns to the original operating point. On the other hand, if a(d P ) / aG is negative, as shown in the central portion of Fig. then an increase in flow decreases the inlet pressure so that the supply flow increases and continues to increase. This flow excursion can be predicted by the Ledinegg criterion,21 8 which states that a necessary condition for the excursion is a(d P ) / aG < where the steady-state characteristic of channel G (d P ) is not single valued with identical inlet enthalpy and heat addition. The flow can be stabilized in this region by providing a pump-supply curve whose slope is even more negative than that of the boiler (right side of Fig. If boiler flow increases, the feed system cannot supply the high pressure needed and flow returns to the operating point. The system is then statically stable even though a(d P ) / aG is negative. Therefore, the criterion for Ledinegg or static stability can be written as
W, 3.23,
0,
3.24).
346
T H E RMAL A N A L Y S I S O F PWRS
a(fl P ) feed aG
a(fl P )]oo p
-----'- < 0 aG
(3.204)
where (fl P )feed is the driving head supplied by the feed pump and (fl P ) ] oo p is the flow friction head demanded by the loop. A system that is statically unstable can exhibit flow oscillations, rather than a one-time excursion, if there is sufficient interaction and delayed feed back between the flow rate and mass accumulation (i.e., compressibility) in the heated section. Stenning and Veziroglu2 1Y call this type of oscillation pressu re drop oscillation. It can generally be avoided by increasing the pres sure drop at the channel inlet or by using a pump with a large head re duction at high flow rates. The techniques used for the analysis of dynamic instability will also determine whether pressure drop instability exists. 3 .4. 1 .3
Dynamic Instability
A temporary reduction in the inlet flow to a boiling channel increases the rate of evaporation, thereby raising the average void fraction. This distur bance affects elevation, acceleration, and frictional pressure drop, as well as heat transfer behavior. For certain conditions of channel geometry, ther mal properties of the heated wall, flow rate, inlet enthalpy, and heat flux, resonance with sustained oscillation can occur. Stenning and Veziroglu 21 Y found that the frequency of a density wave oscillation is higher than that of a pressure drop oscillation (Fig. Neal and Zivi220 recognized that this oscillation is caused by flow-void feedback when a 180-deg phase shift exists between a flow disturbance and its void volume response. This can occur even when the boiler is opera ting in a region where a(fl P ) aG is positive. We consider a single, closed loop containing pump, boiler tube with constant heat input, and condenser and assume that some sinusoidal flow oscillation at a specified frequency is introduced. If pressure drop across the boiler is examined, pressure oscillations are observed which are not necessarily in-phase with the flow oscillation; they can lag the flow oscilla tions. We assume that this process is repeated for a number of different im posed frequencies. At each frequency, the lag (phase angle) between the velocity and pressure oscillations is determined as well as the ratio of am plitude of flow oscillation A F to the a mplitude of the pressure oscillation, Ap . A polar coordinate plot, usually called a Bode diagram, of a typical set of results is shown in Fig. (Ref. At steady state (zero frequency), pressure and flow are in phase, and the boiler has a completely positive resistance. Where the curve crosses the horizontal axis again, pressure and velocity are out-of-phase by 1 80 deg, and the boiler has a completely neg ative resistance. To determine whether instability will occur in our system, we must per form a similar experiment with our feed system and expect to get results
3.25).
/
3.26
217).
H Y D RO D Y NAM I C S
26
347
D 0 . 1 475 i n . T in 6 7 ° F; Pex i t (Freon - 1 1 ) Length/D 250 =
=
24
=
22. 'in a.
.. �
<J
20 .
Density Waves
�
f� 0 .33 cps
�,
I
1 8.
, '"
,
,
,
Pressure Drop Osc .
(�
Fig.
o
3.25
/ �
"'Q
f � . 0 .024 c ps 1 6.
1 5 psia
=
1 .0
3 .0
2.0
Mass F l ow Rate , I b/m i n
Density and pressure wave oscillation regions [from Proc. 1 965 Heat Transfer alld Stanford University Press, Stanford, CA (1965)].
Fluid Mechanics Institllte,
1 .0
NEGATIVE RESI STANCE
0.
5
STEAD Y STATE"
POS ITIVE R E S I STA NCE
0.2 Fig. 3.26
Boiler-inlet impedance i n polar coordinate form [from Ref.
2 1 7] .
T H E RMAL A N A L Y S I S OF PWRS
348
1 0
I
I
rFE E D S Y S TEM
5 1 .2 0 .5 ----' '"----" 0 . 2 .
NEGATIVE R E S I STAN C E
POS ITIVE R E S I STAN C E
0 . 005 Fig. 3 . 2 7
Boiler-inlet a nd
feed water
impedance [from
Ref.
2171.
such as those in Fig. 3.27 (Ref. 21 7) . The feed system has a positive resis tance at all frequencies. For the frequency range where the boiler has a negative resistance, R B , the feed system has a positive resis tance, RF As long as RF > R B , the system is stable. If this is not the case, dynamic insta bility results. -
3 .4 . 1 .4
Parallel Channel Instability
Parallel channel instability can arise when a number of channels are con nected at common headers. Under these conditions, total flow can remain constant, but oscillations in the flow to some channels can occur. This prob lem is of particular concern in a pressure tube reactor where parallel chan nels are connected only at the inlet and exit. The problem can also arise in once-through steam generators. Moxon22 1 notes that parallel channel insta bility is seen in such generators when a pressure drop perturbation arising from the evaporator and superheater is roughly equal to and 180-deg out of phase with a perturbation originating at the inlet. 3.4. 1 .5
Thermal Oscillation
This oscillation occurs in a film boiling region during rapid heating of a cryogenic system. It is a thermal response of the vapor film to flow distur bances. Disturbances that can be reinforced by a mechanism such as acous tic resonance become dominant in the system; these oscillations are often called acoustic instabilities. The velocity of sound is greatly reduced by the
H Y D RO DY NAM I CS
349
presence of small quantities of void; this can lead to "pipe organ" reso nances in short pipe lengths. Observed frequencies have been related to the harmonics of a pipe that is open at both ends, and to Helmholtz resonance. This phenomenon has been reported by Edeskutz and Thursten.222 The inception of oscillations can be predicted by (3.205) where Hfg is enthalpy change on vaporization and the range of (v,' /vr ) is 9 to 25. Cornelius and Parker223 detected acoustic instability (5 to 30 cps) of re frigerant Freon-1 1 4 in a natural circulation loop at a bulk temperature be low critical temperature, but at critical pressure. Walker and Harden224 suggested that the region of pressure and flow fluctuations associated with a natural circulation loop operating at supercritical pressures occurs near maximum in a plot of density enthalpy product versus fluid temperature. Bergles et al. 225 examined thermal oscillation in boiling water flow at 1000 psia. They found oscillations characterized by a frequency > 35 cps and most pronounced in the subcooled region. They also found pressure-drop amplitudes very large compared to steady-state values. The oscillations are considered to be acoustic in nature and induced by the collapse of sub cooled voids. 3.4. 1 .6
Ch ugging
The term ch ugging is usually reserved for the cyclical phenomenon char acterized by the periodic expulsion of coolant from a flow channel. The effects range from simple transitory variations of the inlet and outlet flow rates to a violent ejection of large amounts of coolant, usually through both ends of the channel. The chugging cycle consists of incubation, nucleation, expulsion, and reentry of the fluid. In the double-ended-expulsion case, when the two liquid slugs entering the channel collide, the vapor phase probably collapses. Although chugging was used by Fleck226 to describe gross instabilities in the SPERT-1 reactor system in 1 959, little evidence of similar phenomena has been observed since that time. Direct condensation chugging has been observed in BWRs where steam is directly vented into a water pool so as to limit the pressure rise in the containment vessel. At low steam flow rates (or at higher flow rates with low pool temperatures), water is forced out of the downcomer (injection pipe) and a steam bubble grows at the end of the downcomer. The bubble then condenses and water surges up the downcomer in waterhammer fash ion. Since vapor suppression pools have not been used in PWR design, this phenomenon is not a problem for the PWR designer.
350
TH E RMAL A N A L Y S I S O F PWRS
Effects of Various Parameters on Flow I nstab i l ity
3 .4.2
The effects of various flow-instability parameters, observed in studies pre viously mentioned, are as follows:
1. 2. 3. 4.
5. 6.
High heat input aggravates flow instability because of the resulting high rate of vaporization. Increased inlet subcooling within a critical value reduces stability. Be yond that value, a further increase in subcooling can reverse the trend and stabilize flow. Increased resistance at the exit strongly decreases stability since it in creases exit pressure drop when a large void is generated in the channel. Increased resistance at the inlet strongly increases stability. When flow is slowed by a disturbance, a large pressure head is retained at the inlet and is available for forcing the fluid through the channel, thereby stabilizing the flow. Increased system pressure increases stability in proportion to the in crease in reduced pressure, primarily because the difference in specific volumes of vapor and liquid then becomes smaller. Increased static head in a long vertical channel increases the stability of upward boiling flow and decreases the stability of downward boil ing flow. This occurs because the buoyancy of the vapor aids upward flow and opposes downward flow. With the latter, there is a tendency for the vapor to agglomerate and remain in the channel.
Acoustic or thermal oscillations are primarily associated with startup and transient situations220 and are usually not the major concern of the designer. Flow-pattern instabilities are imperfectly understood . Experimental obser vations are recommended. At present, the designer directs his major con sideration to parallel channel instability and to Ledinegg and dynamic instability. Analysis for the last two problems can be carried on simulta neously since Ledinegg instability can be considered a special dynamic case occurring at zero frequency. These problems are fairly well understood and a theoretical analysis can be made.
3 .4.3 3.4.3.1
Dynamic I nstability Analysis Basic Thermohydrodynamic Equations
Tong89 reviewed this area extensively; we follow his discussion. First, we write the conservation equations for one-dimensional flow, with slip, in a vertical channel. We slightly modify equations and for the mass balance to obtain
(3. 1 25), (3. 1 26), (3.1 27)
H Y D R O DYNAMICS
(3.206)
z
where indicates axial distance and VI and Vv are the liquid and vapor velocities, respectively. For the energy balance, we have
i [ P / (1 - a)HI + p"aHv ] + � [ p/(1 - a)VIHI + Pv aV" Hv ]
at
az
q lll
= q lll
(3.207)
where energy dissipation is neglected, is the heat input per unit volume of fluid and HI and H" are the liquid and vapor enthalpies, respectively. The foregoing assumes operation at a nearly constant pressure so that dH = The momentum balance can be written as
dE.
±
[ P/(1 - a) + pva] gc� = 0
(3.208)
A positive sign for the final term indicates upward flow; a negative sign indicates downward flow; F is the frictional force per unit volume of fluid. These conservation equations assume that mixture behavior is ade quately defined by overall conservation equations obtained from summing the conservation equations for the individual phases. Furthermore, each individual phase is assumed to occupy a given fraction of the channel cross section and to have a velocity that is constant across the cross section occupied. * If the system remains at a nearly constant pressure, the equation-of-state can be written as
(3.209a) (3.209b) Note that in subcooled boiling, the mixture density should be calculated on the basis of the amount of local boiling void present (see Sec. 3.3.3.2). The preceding equations can be simplified by introducing the quality defined as
(3.210)
*Two-phase flow conservation equations have been written in a variety of forms. For a more comprehensive view of this subject, the reader is referred to Refs. 227, 228, and Sec. 3.3.4.
T H E RMAL A NA L Y S I S OF PWR S
If local mass velocity is defined as
(3.2 11)
G apvVv + 0 -a)PIVI ap aGz -+-=0 at a =
the continuity balance becomes
The volume-weighted mean density, p, is given by Eq. balance is given by
(3.75)
_aH P at -Hfg aat G aH_ az - q PI X 0 -a) - PvaO .- X) �
The quantity � defirned by
+
(3.212) and the energy
(3.213)
'"
(3.214)
�=
H
is a slip correction !for the energy balance and is zero for nonslip flow. Mixing cup enthalpy is defined by
H=HI + X(Hv-H/) �gc [ aGat + �az (V I G 2 ) ] apaz +F ± Pggc v I effective specific volume for spatial acceleration, X 2 - )2 v ' = 2G1 [PIV?0 -a) + pVVl� a] =-+ apv (0I - aX)PI v ' = 1/ p = p (H )
(3.2 15)
The momentum balance is
+
where
the
This becomes written as
(3.216)
=0
is
(3.217)
P for monslip flow, and the equation-of-state can be
(3.218)
If channel size is small (e.g., between fuel rods in a PWR core) and vapor quality is sufficiently low so that bubbly flow exists, slip is negligible at high pressures. This is particularly true during a flow oscillation, since void distribution changes as flow changes from accelerating to decelerating. 3.4.3.2
L inearization and Laplace Transformation (Frequency-Domain Ana lysis)
When we consider a steam generator consisting of a large number of boiler tubes, each generating approximately the same amount of steam, it is usu ally adequate to consider the steam generator as a single channel. If we
H Y D RO D Y N AMICS
(3.207), (3.208)
utilize the technique of small perturbations, the variables in Eqs. and can be described by
G (z,t) = G (z,O) + dG(z,t) H(z,t) =H(z,O) + dH(z,t) p (z,t) = p (z,O) + d p (Z,t) q '" (z,t) q '" (z,O) + dq '" (z,t) dG(z,t) / G(z,O)« 1, =
(3.206), (3.2 19)
where etc. If these variables are substituted into the conservation equations and if the second-order terms are ignored, the equa tions can be linearized to yield
adGaz ad p at adH A C -iJH +p -adH uA q '" = G --+u az dz at
(3.220) (3.221)
� p = .E£.aH�H+ aGap dC+ � iJq '" �q '"
(3.223)
Assume that we wish to examine the system stability with respect to inlet mass flux perturbations. In the usual approach, the linearized conti nuity, energy, and density equations [Eq. and are integrated to obtain the time-dependent distribution of density, mass flux, and enthalpy along the channel. The integration assumes that properties are independent of pressure and hence the continuity, energy, and density expressions are decoupled from the momentum equation. The result of the integration is then used in the momentum equation to calculate the pressure drop perturbation due to an inlet mass flux perturbation. The Laplace trans formation of the resultant equation is then obtained. Recall that the Laplace transform, 5£, of a function of time, is given by
(3.220), (3.221), (3.223)]
(u + j
t(t), (3.224)
where s is a complex number w). The real part, (J', gives the amplitude coefficient whereas the imaginary part, w, represents the angular frequency of oscillation. The Laplace transform technique converts a partial differen tial equation into an algebraic equation in terms of s. The result of the
T H E RMAL A NA L Y S I S OF PWRS
3 54
transformation yields a characteristic equation H (s), which represents the dynamic response of the system in the frequency domain. That is,
= d= ,
= O(P) dPexit esf o( P )
o(G) /H (s)
(3.225)
where H (s) open loop transfer function expressed as an algebraic equation = in s, o(G) Ge t, and The stability o f the channel i s deter mined by setting H (s) equal to zero and finding the roots of the equation. The system is stable only if all the roots have negative real parts. For the stability analysis of an entire system, rather than just a boiling channel, the behavior of all the system components must be modeled. If the system can be modeled as a sequential set of components in a single loop, the system transfer function, C(s), is then determined from C (s)
=L
Hi (s)
(3.226) (3.226)
where Hi (s) is the transfer function for component i. Equation re quires that all components be approximated by linear differential equations with constant coefficients. Expressions for the transfer functions of a variety of components are available in the literature. 229 In determining system sta bility, the same criterion is applied to C (s) as previously applied to H (s). Equation may also be used to avoid the difficulties of analytic integration along a boiling channel. The channel, or the entire system, may be divided into i subsections in which all variables and parameters are treated as unchanged and having their average steady-state value over the node. The equation set for each node is perturbed, linearized, and Laplace transformed to obtain the transfer function for each node. The transfer func tion for the channel or system is then the sum of the transfer functions for the individual nodes. Linearized models using a frequency-domain analysis, such as exempli fied by the STABLE program230 have been shown23 1 to be capable of providing reasonable predictions of existing data. 2 19,23 1 -235 A more recent example of the frequency domain approach is the DYNAM/ US236 program.* It is possible to apply the frequency-domain approach to systems con taining a number of parallel channels. Lahey237 indicates how a nodal ap proach with an appropriately defined transfer function may be used. The DENSITY-PARA program238 provides an example of the application of this approach.
(3.226)
3
*Those readers interested in frequency-domain stability analyses, which include reactivity feedback effects, should see Chapter 7 of The Thermal-Hydraulics of a Boilillg Water Nuclear Reactor, 2nd. ed, by R. T. Lahey, Jr., and F. J. Moody, American Nuclear Society, LaGrange Park, IL (1 993).
H Y D RO D Y N AM ICS
3 .4.3 .3
Non linear Techn iques
The chief advantage of linearized approaches is that perturbation of the governing equations around the steady-state operating conditions and con version of the resulting linearized model to the frequency domain yields analytical solutions for the system transfer functions. These functions can then be examined using well-established criteria to obtain rigorous condi tions for system stability. The approach is useful in determining 0 ) whether a particular operating point is stable and the conditions needed for the onset of instability. However, the approach does not determine the range of stability for conditionally stable systems (system is stable for infinitesimal perturbation but unstable with a finite perturbation of sufficient magnitude) or the responses of an unstable system. In addition, the linear response assumed may not be a very good approximation of actual system behavior. Nonlinear approaches to dynamic instability have been used by a num ber of investigators (e.g., Refs. through The basic approach first divides the system examined into a number of control volumes. The con servation equations are then written for each control volume in finite dif ference form. Steady-state conditions are assumed and the conservation equations are numerically integrated along the channel or channels. One of the system parameters (Le., pressure, inlet flow, heat flux) is subjected to a finite-sized perturbation. The time variation of system behavior is then de termined by numerically integrating the conservation equations along the channel for a series of time steps. If the perturbation produces oscillations in flow rate that damp out with time, the system is stable; if the oscillations persist, the system is unstable. Rosa and Podowski241 point out that care must be taken to assure that an adequate number of axial nodes is provided. Failure to do this can lead to misleading results.
(2)
239
3.4.3.4
242).
Simplified Criterion for Dynamic Instability
It is desirable to be able to obtain a preliminary estimate of the dynamic stability boundary under many circumstances. If the boiling channel can be considered to be placed between two plenums across which a constant pres sure drop is maintained (equivalent to a feed pump with a flat characteristic in the range of interest), Ishii's243 stability criterion can be used. At high inlet subcooling, he concluded that the system is stable when Npch < """-" Nsub +
where
2[ Ki ( fm 2 ;;) Ke 1 21 [( fm 2 + 2Kcl +
+
/ D +
1
/ DV
Npch = equilibrium phase change number = !1cq L / Vi
!1cq = equilibrium frequency of phase change, rad /s = rg � p / (pg P/ )
(3.227)
T H E RMAL A N A L Y S I S O F PWRS
356
mass rate of vapor generation per unit vol, mass/vol s Nsub = subcooling number = ( � p / Pg )(� Hsub / � Hfg ) � Hsub = inlet subcooli ng = Hsat - Hin � Hfg = enthalpy change on evaporation Hin, Hsa t = enthalpy of liquid at inlet and saturation, respectively Pg , PI = gas and liquid densities, respectively rg =
�P
1m Ki,Kc Vi
=
Dh
L
= =
P/ - Pg
two-phase mixture friction factor hydraulic diameter, Dj; = D'l / L length of heated channel inlet and exit orifice coefficient, � p = KC 2 / pgc fluid velocity at channel inlet.
The stability criterion of the previous equation holds only if Nsub > ( Nsub )cr and ( Nsub )cr is obtained from244
(3.228) (3.229)
where Pe
=
Peclet number for liquid at inlet conditions
Ac = cross-sectional area of boiling channel
heated perimeter of boiling channel ( Npch )o = value of Npch determined from linearized stability calculations for zero sub cooling at the inlet. Ph
=
1.0.
Under most conditions, ( Npch )o occurs at channel exit quality xe ' which is close to Therefore, an approximate value for ( Npch ) o can be obtained from this fact and the relationship
(3.230)
At inlet subcooling numbers below ( Nsub ) cr , the system is expected to be stable if (N ch p
) :s;
[
]
( Npch )cr - ( NpCh ) O Nsub + ( Npch ) ( Nsub ) cr
0
(3.227)
(3.231)
when Nsub = where ( Npch )cr = the value of Npch obtained from Eq. ( Nsub) cr ' The Freon loop stability data of Saha et a1. 244 appear to be in good agreement with this simplified approach. 3 .4.4
Parallel Channel I nstabil ity
Momentum integration is particularly useful in treating the problem of par allel channel instability. Nonlinear transient momentum equations can be
H Y D RO D Y N AM I CS
conveniently solved for several channels by integrating the momentum equation along each channel. Fluid properties are obtained from the energy equation and the equation-of-state; integrated equations are then solved simultaneously. These techniques have been incorporated into the HYDNA computer code,245 which predicts flow oscillation in parallel channels. To describe the HYDNA computational method, we again follow Tong.89 It is useful to separate inlet mass velocity and the instantaneous axial mass flow rate distribution. If is the axial mass flow rate distribution in the channel and is the inlet mass flow rate, which can vary with time, we have
G(O,t) (z,t) G (z,t) = G (O,t)(z,t) (3.232) G (z,t) z t, f LFdz + f L ± pdz+ -1 Ll(v ' G 2 )+ -1 G(O,t)f L-dz (l LlP+ 0 0 at gc gcL dG(O,t) - - o dt -g1cf0 dz (3.233) v' = 1/ pLlP J5 dz, a / at G dG(O,t)/dt LlP,G (z) C B
Substituting this, where is mass velocity as a function of and the momentum balance over the channel length gives
into
---
-------
where
is the pressure difference between the common plenums and for homogeneous flow. Factor which accounts for the effect of axial variation of the mass velocity, comes close to unity for a slow transient if remains nearly constant throughout the passage. However, vanishes if the shape of remains constant. All terms of Eq. (3.237), except and can be easily determined. Let and be given as
(3.234) C= - �f L1 gc 0 dZ f LFdz + f L ± pdz + �d(V ' G 2 ) + �G(o,t)f La dz 0 0 at gc gc (3.235) B= - o �g f LdZ c ---
and
-------
0
Substitution into the momentum balance yields
dG(O,t) = CAu P+B dt
--
(3.236)
T H E RMAL A NA L Y S I S O F PWR S
358
Equation
(3.236)
can be written for each of
11
parallel channels
dGl (O,t) C} LlP + B} dt dG2 (O,t) C LlP + B 2 2 dt dGn (O,t) C Ll P + B dt With a given total inlet flow rate wet), A 1 GI (O,t) + A2 G2 (O,t) + + All Gil (O,t) = W(t) where A I ' A2 , , An are the respective flow areas of channels 1/
Differentiation of Eq.
(3.238)
(3.237)
1/
gives
1, 2,
(3.238) 11 .
+ A 11 dGndt(O,t) dW dt or
(3.239) The latter equation can be used to evaluate the pressure drop that is com mon to all channels; i.e.,
(3.240) (3.237).
(3.239)
Equation can be used to predict the pressure drop as a function of time by iterating on it along with 11 simultaneous equations of Eq. The steps in the calculation for a small time increment are given by Tong89 with a method that can predict the thermal and hydraulic behavior of the system in response to a specified variable inlet flow rate or a given variable power input to the channel. The general nonlinear approach can also be used for parallel channels that are interconnected . Of course, the conservation equations for each channel modeled must be appropriately altered to allow for interchange between the channels if an interconnection is assumed. The PARALLEL246 and LOOp22I codes are examples of such alternatives. The PARALLEL pro gram uses the nonlinear technique described in Sec. but provision for parallel boiling channels is included.
3.4.3.3
H Y D R O DYNAMICS
3 .4. 5
359
Flow Pattern I nstabil ity
An instability that appears to fall into this classification may be seen in a LOCA. During the later stages of blowdown, refill of the lower plenum, and reflood of the core, the subcooled emergency core coolant (ECC) may interact with steam flowing in the cold legs to produce flow and pressure oscillations or rapid transients. Oscillations have been observed in scale model tests (1 / 30 to 1 /3 PWR scale) of the ECC injection region. It appears tha t this is primarily a flow pattern transition instability due to the change from dispersed liquid to a liquid slug near the injection point. The heat transfer rate to droplets is high with the result that the pressure is reduced. A plug of injected liquid then moves upstream and covers the injection port. The condensation rate is reduced and the pressure increases so that the steam from the steam generator flows into the injection area. Capiello and Liles247 were able to predict the oscillation in ECC flow observed in a 1 / 14-scale experiment. By using a version of TRAC that con tained a stratified-slug criterion in the code's generalized flow map, oscil latory flow behavior was indicated. The predicted oscillation frequency was, however, somewhat lower than observed. 3.4.6
PWR I nstabil ity Concerns
Under the conditions that may be expected during anticipated transients and normal operation of vessel-type PWRs, no instabilities are observed in the primary system. There is relatively little boiling in the core and hence little void reactivity feedback. Further, all the core channels are intercon nected. It has been shown, both experimentally and analytically, that a set of interconnected channels will have a much larger area of stable operation than a set of isolated parallel channels. 248 The situation in PWR cores is in marked contrast to that seen in BWRs. In a BWR, the coolant channels are isolated and the large void fraction in the core leads to a substantial void-power feedback. BWRs therefore operate closer to the dynamic instability threshold. Under natural circulation con ditions, flow and neutron flux instability have actually been observed in some operating reactors. 249 Instability in pressure tube PWRs is more of a concern than in vessel type PWRs. Such systems must be carefully analyzed to determine whether parallel channel instability presents a problem. The situation is of most concern in some newer CANDU designs where boiling in the core at high power is utilized giving an average outlet quality of up to 4%. 250 This leads to consi derably greater void fraction in the core than in the usual vessel type PWR. Void reactivity feedback can no longer be neglected. To improve the stability of the new CANDU designs, some system mod ifications have been made. As usual, the heat transfer system is arranged in two essentially independent circuits; one on each side of the vertical axis
TH E RMAL A NA L Y S I S OF PWR S
3 60
STEAM GENERATOR 2
LOOP
I
REACTOR LOOP 2
ECcc_REACTOR OUTLET INTERCONN= T __ __ --.------
Fig.
3 .2 8
Use of reactor outlet interconnect to enhance CANDU stability [from Garland and Pang, Nuc!. Technol., 75, 241 (1 986)] .
3.28).
of the core. Each circuit contains two inlet headers, two steam generators, and two outlet headers (see Fig. The new systems are modified by an interconnection line, which connects the outlet headers of each circuit. The system stability analysis shows the system to be stable during normal operation with the interconnection line in place but that parallel channel instabilities would occur if it were removed. If a flow perturbation occurs, the interconnection reduoes the effect of the perturbation by allowing fluid to move from the remote header to the perturbed header. The ability of the fluid to move frorm the perturbed header is enhanced if the interconnect contains vapor. To increase the fraction of steam in the interconnect line, the line is elevated and drainage is provided. Flow stability is also a concern in the Savannah River Site (SRS) pro duction reactors. 'J1hese vertical pressure tube reactors are more susceptible to flow instability than power reactors because of the low pressure ba r) at which SRS reactors operate. While SRS reactors are stable at normal operating conditions, local flow excursions are possible under accident con ditions.251 During a postulated accident in which reactivity is inserted in one flow channel, the increased power would produce substantial boiling and void production. The shar!ply increased channel pressure drop leads to
(2
H Y D RO D Y N AM ICS
361
a situation where the Ledinegg stability criterion is not met. The channel flow is then sharply reduced, leading to inadequate cooling of the fuel. The channel power level at which the flow excursion occurs is lower than the power level at which conventional correlations predict the occurrence of the critical heat flux (see Sec. for critical heat flux discussion). The margin to the onset of flow instability is therefore the limiting factor in setting the reactor operating power. In addition to analyzing the primary system, a stability analysis of the steam generator should also be conducted. Unstable operation may be pos sible under some transient or accident conditions. Flow oscillations during a LOCA are also of some concern. The flow pattern instability possible in the ECCS injection line has been described in Sec. Flow oscillations have also been observed during the bottom reflooding of a core after a LOCA. 252 White and Duffey253 have proposed a simple model in which the liquid in the downcomer and the two-phase mixture around the fuel are regarded as a piston trapping a volume of steam, which is leaking out through an outlet orifice. Their model predicted oscillation frequencies that depended on the steam generation rate and that were in rough agreement with experiment. Since the steam generation dy namics depend strongly on the quench front, this is a unique phenomenon.
4.42
3.4.5.
3.4.7
Recommended Methods of Analysis
The onset of static instability in a natural or forced circulation loop can be determined by the Ledinegg criterion21 8 with a lumped pressure drop. The avoidance of dynamic and parallel channel instability remains the major concern of the designer. The early preference for the linearization and Laplace transform approach for analysis of the dynamic instability was clearly based on the relatively short computer times required. However, with the very large increase in computer power that has become available, the longer computing times of the nonlinear approach are now not much of a problem. The nonlinear techniques allow a better representation of the problem as well as the examination of conditional stability. Further, most of the widely used accident analysis programs can be easily adapted to stability analysis. 25o,254 Momentum integration remains a very convenient method for examining parallel channel instabilities. However, a conservative design philosophy would dictate that the final (reference design) stability analysis should in clUde the effect of reactivity feedback (also true for vessel cores even though such effects are likely to be quite small) . It may then be easier to use a nonlinear approach. The onset of flow pattern instabilities cannot be adequately predicted in the general case. While the instability seen in the cold leg during emergency coolant injection was predicted after the fact, it was not expected prior to
T H E RMAL A NA L Y S I S OF PWRS
362
the operation of the experiment in which it was observed. Experimental studies are recommended. 3.5
L I Q U I D-VAPOR SEPARAT I O N
Liquid entrainment during evaporation, vapor carry-under in a natural cir culation loop, and steam separation are closely related because they are all controlled by buoyancy forces. Although they are usually considered to be in the province of the boiling water reactor, they must be understood for steam generator design and system behavior during a LOCA. Although not discussed here, note that entrainment phenomena can be significant in se vere accidents. Aerosol generation and molten cladding and fuel relocation models include entrainment as a transport mechanism. Entrainment and Carry-Over
3.5.1 3.5.1 .1
Entra in ment and Carry-Over from Horizonta l Surfaces
Early studies of entraiment from a horizontal liquid surface were carried out by Sterman/55 Newitt et af. 256 and Mitsuishi et al. 257 Mitsuishi et al. noted that at low gas superficial velocities most droplets injected into the vapor stream fell back into the liquid and only the smallest droplets were carried off by the vapor. However, above a superficial velocity of ftl s at atmospheric pr.essure, entrainment sharply increased and even small droplets were carried along with the vapor. Sterman found that in small size vessels the amount of entrainment depends on the vessel size. Ster man255 provides a:n expression for determining minimum diameter at which entrainment is independent of vessel size: [Pv I (Pl - Pv )] - 0. 2
�6
D
[ a ' I (Pl - p t, ) ] 1/2
2:
-
260
D (3.241)
where
a ' = surface tension, lb I ft PI = liquid density, Ib / fP Pv = vapor density, Ib/ ft3
D
=
diameter, ft.
We may define carry-over as those entrained liquid droplets which are carried along by the vapor and do not fall back into the liquid. Davis258 examined water entrainment and carry-over at high pressures. He found that the maximum height !to which droplets are carried by rising steam increases with increasing steam velocity. This is clearly seen in the inser t of Fig. The height at which the water droplets are observed gradually increases until a critical carry-over velocity is reached and surface separa-
3.29.
H Y D RO D YNAM I CS
1 6 r-------------
363
-.�----------------------�
______
� 24 �----�--r---��
Q.
Q.
o
I
(5 1 6
14
o
12
......
o
':--.. .!:...
Ol
Q) I
10
�
� u o Q)
> ...
8
o o M
a...
o o -0
:c
-'
on
. o;;;
X o
8
O L...--.....L----'--..!..oi 3.0 1 .0 2.0 Ve l oc i ty of R i s i ng Steam, his
Q) >
� 6 C
o I u
4
2
400
800 Satura t i on Pressure, ps ia
Fig. 3 . 2 9
Surface disengagement carry-over velocities. Notation: Curves from Ref. 258 + Allis Chalmers, 33-in. sep. ht. L,. General Electric, 32-in., sep. ht. • Combustion Engineering, 50-in. sep. ht.
o •
•
Combustion Engineering, 3D-in. sep. ht. Combustion Engineering, 24-in. sep. ht. N. Mitsuishi et ai., AEC-TR-4225, US. Atomic Energy Commission, Washington, D.C. (1961 ).
tion can no longer be achieved. At high pressures, the carry-over occurs more suddenly and at a much lower steam velocity. The carry-over velocity computed by Davis as a function of pressure is shown in Fig. Also shown are data obtained in various U.s . government-sponsored programs. Note that the data are in reasonable agreement with the Davis curve. The Davis curve applies to carry-over from large size vessels. Wilson and M cDermott's259 data can be used in evaluating carry-over in small flow cha nnels with cold walls.
3.29.
T H E RMAL A NA L Y S I S OF PWRS
3 64
As indicated in Chapter 1, surface separation is used in the horizontal steam generators included in the Russian VVER system. All of these steam generators contain a horizontal perforated plate located slightly below the normal liquid level. It is found that such a plate substantially reduces the entrainment rate. 260 Significant carry-over would be encountered during the reflood phase of a hypothetical LOCA. iIn this period, the core would be partly refilled, but there would be substamtial steam generation due to the decay heat released by the fuel rods. When the average steam velocity exceeds the carry-over velocity (Fig. 3.29), a portion of the water being added to reflood the core is removed by entrainment. The amount of this entrainment is generally e�timated from empirical correlations of test data. The rate of entrainment, ME ' is sometimes given in terms of the carry-over fraction. That is, . . . ME = (CRF)(Mf ) - Msg
(3.242)
where Mf = rate at whicln. water is added to core, mass /time C�F = carry-over fraction Msg = rate of steam generation in the core, mass /time. The carry-over fraction (rate of total core exit flow rate to core inlet flow rate) was correlated bf6 1 CRF = 0.9553 exp[ - 0.0013935( P / 60) 2 ] [1 - exp[ - 4.31 27(q ' / 1 .24)] X
{I
- exp[ - 16.62(V /6)]} exp[ - 0.0371 61 (Ll T / 140) 2 ]
1 + sin3.l41 6[ - 0.5 (Z/ 6)0 . 025733 (6/V ) + 0 . 13553( tlT / 1 40) + O .0656( P / 60) ] X ------
+
2
(3.243)
where P = pressure, psia q ' = peak linear heat rate, kW / ft = flooding rate, in. / s Ll T = coolant subco0ling at inlet, OF Z = quench front level, ft.
V
Note that Eq. (3.243) is an empirical correlation of data for PWR ECCS evaluation during a hypothetical LOCA. It should not be used for other conditions. Entrainment ra tes from horizontal surfaces are also of concern to the process industries and more general correlations are available from this
H Y D R O DYNAMICS
365
source. Azbel and Liapes262 provide an entrainment rate correlation in terms of dimensionless groups. Based on data from several systems, they conclude: (3.244) where
Ge = superficial mass flux of entrained liquid, mass/area time Gg = superficial mass flux of vapor, mass/ area time
Vg�Pg Vf� g2[IT'/(gLlp)3j2Ls (Ll p) ' = L, : ( g p
A =
------
-----
B [ J/ I =
surface tension
=
available height above separating surface
VI = kinematic viscosity of liquid Llp = Pl - Pg Vg = superficial velocity of vapor IT
L5
and other symbols have their previous meaning. Note that Eq. (3.244) would predict a near zero carry-over if a long disengagement height is provided. 3.5 . 1 .2
Entra inment in A n nular Flow
In adiabatic annular flow, liquid droplets are continuously entrained from the liquid film on the tube surface. At the same time, some of the liquid droplets suspended in the vapor core are deposited on the liquid film. For long unheated tubes, the rate of deposition is in equilibrium with the rate of entrainment, and the film thickness remains constant. Modeling of annular flow is usually based on procedures developed at Harwell. 263 The deposition rate, (mass / area time), is determined from the product of a mass transfer or deposition coefficient Kd and droplet concen tration C; that is,
R
(3.24Sa) where C is determined from (3.24Sb) where WE = mass flow rate of entrained liquid WG = mass flow rate of gas
T H E RMAL A N A L Y S I S O F PWR S
366
Pg PI
= =
gas density liquid density.
Kd
Values of are based on correlations of experimental measurements. The entrainment rate is based on the observation that under equilibrium con ditions during adiabatic operation, the entrainment rate, E ', and deposition rate are equal. Since under these conditions E ' = R and R =
Kd C,
CF
Kd 2 ( CE ) -O. ( ) Kd =0 023V � = CE
(3.246)
where is the entrained droplet concentration under adiabatic equilibrium conditions. An estimate of may be obtained from264
(3.247)
DV
.
�
Pg /-Lg ' where Vg = vapor velocity and Prg gas phase Prandtl number. In the Harwell approach is taken as a function of the dimensionless group (T 8 1 where T = interfacial shear stress, 8 average liquid film thickness, and = surface tension. Calculation of the entrainment rate then requires a knowledge of the interfacial shear stress and film thickness. Use is made of the so-called tri angular relationship, which allows WF,8, and dp/dz to be related in a way that knowledge of any two allows the third to be calculated. The relation ship used is
a ' ) , a'
48 I = 0
where D
= =
I
[ (dp/dz) L F dp/dzl 1/2
=
(3.248)
tube diameter
dp/dz = two-phase pressure gradient (dp/dz)L F pressure gradienlt for single-phase liquid with same mass
flow rate as in liquid film.
The value of (dp/dz) LF can be obtained for any assumed value of the liquid film flow rate, WF . The two-phase pressure gradient is then obtained by determining the pressure gradient produced in the droplet-laden gas core by the friction at the core-liquid film interface. The interfacial shear stress is calculated using the Wallis264 relationship between film thickness and interfacial friction defined on the basis of a homogeneous gas core. That is,
(3.249)
H Y D RO D Y NAMICS
367
where
Ph = homogeneous density of vapor-droplet core, mass /vol + 3008 / 0) =
Cd
f
=
f/4(1
single-phase friction factor based on mass flux and density of vapor-liquid core.
I
Instead of correlating the entrainment rate, E with CE , Sugawara265 correlated it directly with three dimensionless groups. Based on available data, he obtained (3.250) where
I
E = entrainment rate, kg/s m 2
dZ =
height of waves on liquid film,
1 0.578 21.73 10382
and for Reg > dZ =
X
1 05
10683 +
10984
- 38.8 X 55.68 X mean liquid film thickness, m Reg = Reynolds number based on vapor properties and flow rate. Below a critical vapor velocity in annular flow, the entrainment rate is negligible. A simple criterion for the absence of entrainment,266 which is consistent with the previous annular flow entrainment rate discussion, is that the dimensionless group The values of and for the condition of zero entrainment may be obtained by simultaneous solution of Eqs. (3.248) and (3.249) with the assumption that all liquid is in the liquid film. If the foregoing inequality is satisfied with the and values so ob tained, then entrainment rate calculations are unnecessary. When entrained droplets are produced from the liquid film, the droplets will have a range of sizes. Those droplets larger than a critical size are subject to further breakup as they interact with the gas core. The droplet diameter at which breakup begins267 is determined by the critical Weber number, WeC / where (3.251)
8
=
+
X
(T8/0" )�0.01.
dd
Vr
T 8 T 8
is the critical droplet diameter and is the relative velocity between droplet and vapor. Values of Wee for room-temperature water are in the range of 12 to 22 (Ref. 267). The mean drop size can also be expressed in terms of the Weber number. For LOCA calculations, it has been proposed268 that the mean drop size be
T H E RMAL A N A L Y S I S OF PWRS
3 68
estimated using a Weber number of 2.7 Whenever Vr falls below the av erage fluid velocity, [a Vg a)V, ]' the Weber number is calculated usin